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Implications over intuitionistic fuzzy sets: Difference between revisions
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For the various definitions of implication of over [[intuitionistic fuzzy sets]], the functions sg(''x'') and {{overline|sg}}(''x'') have been used: | For the various definitions of implication of over [[intuitionistic fuzzy sets]], the functions sg(''x'') and {{overline|sg}}(''x'') have been used: | ||
<math> | <math>\text{sg}(x) = \begin{cases} 1 \text{ if } x > 0 \\ | ||
\text{sg}(x) = | 0 \text{ if } x \leq 0 | ||
0 | \end{cases},</math> <math> | ||
\end{ | \overline{\text{sg}}(x) = \begin{cases} 0 \text{ if } x > 0 \\ | ||
\overline{\text{sg}}(x) = | 1 \text{ if } x \leq 0 | ||
\end{cases}.</math> | |||
\end{ | |||
<!-- == List of intuitionistic fuzzy implications == | |||
{| width="100%" class="wikitable sortable" style="font-family:Courier; font-size:120%;" | {| width="100%" class="wikitable sortable" style="font-family:Courier; font-size:120%;" | ||
| Line 527: | Line 526: | ||
| | | | ||
| {<x, <font color=green>(1-μ<sub>A</sub>(x)).{{overline|sg}}(1-ν<sub>B</sub>(x))+sg(1-ν<sub>B</sub>(x)){{overline|sg}}(μ<sub>A</sub>(x)+min(1-ν<sub>B</sub>(x),sg(μ<sub>A</sub>(x))))</font>, <font color=red>μ<sub>A</sub>(x).{{overline|sg}}(1-ν<sub>B</sub>(x))+ν<sub>B</sub>(x).sg(1-ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x))</font>>|x ∈ E} | | {<x, <font color=green>(1-μ<sub>A</sub>(x)).{{overline|sg}}(1-ν<sub>B</sub>(x))+sg(1-ν<sub>B</sub>(x)){{overline|sg}}(μ<sub>A</sub>(x)+min(1-ν<sub>B</sub>(x),sg(μ<sub>A</sub>(x))))</font>, <font color=red>μ<sub>A</sub>(x).{{overline|sg}}(1-ν<sub>B</sub>(x))+ν<sub>B</sub>(x).sg(1-ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x))</font>>|x ∈ E} | ||
|- valign="top" | |||
| →<sub>87</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font>, <font color=red>min(sg(ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>88</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),sg(1-μ<sub>A</sub>(x)))</font>, <font color=red>min(sg(ν<sub>B</sub>(x)),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>89</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font>, <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>90</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),{{overline|sg}}(μ<sub>A</sub>(x)))</font>, <font color=red>min(ν<sub>B</sub>(x),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>91</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(μ<sub>B</sub>(x),min(1-μ<sub>B</sub>(x),ν<sub>A</sub>(x)))</font>, <font color=red>1-max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>92</sub> | |||
| | |||
| | |||
| {<x, <font color=green>{{overline|sg}}(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>, <font color=red>min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>93</sub> | |||
| | |||
| | |||
| {<x, <font color=green>1-min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font>, <font color=red>min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>94</sub> | |||
| | |||
| | |||
| {<x, <font color=green>μ<sub>B</sub>(x)+(1-μ<sub>B</sub>(x))<sup>2</sup>.ν<sub>A</sub>(x))</font>, <font color=red>(1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+(1-μ<sub>B</sub>(x))<sup>2</sup>.(1-ν<sub>A</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>95</sub> | |||
| | |||
| | |||
| {<x, <font color=green>min(ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))+sg(μ<sub>B</sub>(x)).({{overline|sg}}(1-ν<sub>A</sub>(x))+min(μ<sub>B</sub>(x),sg(1-ν<sub>A</sub>(x))))</font>, <font color=red>min(1-ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))+min(min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))),sg(1-ν<sub>A</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>96</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font>, <font color=red>min(sg(1-μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x)</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>97</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),sg(ν<sub>A</sub>(x)))</font>, <font color=red>min(sg(1-μ<sub>B</sub>(x)),{{overline|sg}}(ν<sub>A</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>98</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font>, <font color=red>1-max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>99</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-ν<sub>A</sub>(x)))</font>, <font color=red>min(1-μ<sub>B</sub>(x),{{overline|sg}}(ν<sub>A</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>100</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(min(ν<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),μ<sub>B</sub>(x))</font>, <font color=red>min(min(μ<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),ν<sub>B</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>101</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(min(ν<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),min(μ<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font>, <font color=red>min(min(μ<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),min(ν<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>102</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(ν<sub>A</sub>(x),min(μ<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font>, <font color=red>min(μ<sub>A</sub>(x),min(ν<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>103</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(min(1-μ<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),1-ν<sub>B</sub>(x))</font>, <font color=red>min(μ<sub>A</sub>(x),sg(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>104</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(min(1-μ<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),min(1-ν<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font>, <font color=red>min(min(μ<sub>A</sub>(x),sg(1-μ<sub>A</sub>(x))),min(ν<sub>B</sub>(x),sg(1-ν<sub>B</sub>(x))))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>105</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(1-μ<sub>A</sub>(x),min(1-ν<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font>, <font color=red>min(μ<sub>A</sub>(x),min(ν<sub>B</sub>(x),sg(1-ν<sub>B</sub>(x))))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>106</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(min(ν<sub>A</sub>(x),sg(1-ν<sub>A</sub>(x))),μ<sub>B</sub>(x))</font>, <font color=red>min(min(1-ν<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),1-μ<sub>B</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>107</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(min(ν<sub>A</sub>(x),sg(1-ν<sub>A</sub>(x))),min(μ<sub>B</sub>(x),sg(1-μ<sub>B</sub>(x))))</font>, <font color=red>min(min(1-ν<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>108</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(ν<sub>A</sub>(x),min(μ<sub>B</sub>(x),sg(1-μ<sub>B</sub>(x))))</font>, <font color=red>min(1-ν<sub>A</sub>(x),min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>109</sub> | |||
| | |||
| | |||
| {<x, <font color=green>ν<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),μ<sub>B</sub>(x))</font>, <font color=red>μ<sub>A</sub>(x).ν<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>110</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>, <font color=red>min(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>111</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font>, <font color=red>min(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>112</sub> | |||
| | |||
| | |||
| {<x, <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font>, <font color=red>μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)).ν<sub>B</sub>(x)</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>113</sub> | |||
| | |||
| | |||
| {<x, <font color=green>ν<sub>A</sub>(x)+(μ<sub>B</sub>(x).ν<sub>B</sub>(x)-ν<sub>A</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font>, <font color=red>(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x))).(ν<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>114</sub> | |||
| | |||
| | |||
| {<x, <font color=green>1-μ<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font>, <font color=red>μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+min({{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>115</sub> | |||
| | |||
| | |||
| {<x, <font color=green>1-min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>, <font color=red>min(μ<sub>A</sub>(x)(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>116</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(1-μ<sub>A</sub>(x),(1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)+{{overline|sg}}(ν<sub>B</sub>(x)))</font>, <font color=red>min(μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x).((1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)+{{overline|sg}}(ν<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>117</sub> | |||
| | |||
| | |||
| {<x, <font color=green>1-μ<sub>A</sub>(x)-ν<sub>B</sub>(x)+μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font>, <font color=red>(μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x))).ν<sub>B</sub>(x)</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>118</sub> | |||
| | |||
| | |||
| {<x, <font color=green>(1-μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x))+μ<sub>A</sub>(x).ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))</font>, <font color=red>(μ<sub>A</sub>(x)-μ<sub>A</sub>(x)<sup>2</sup>+{{overline|sg}}(1-μ<sub>A</sub>(x))).((1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)<sup>2</sup>+{{overline|sg}}(1-ν<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x))</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>119</sub> | |||
| | |||
| | |||
| {<x, <font color=green>ν<sub>A</sub>(x)+min({{overline|sg}}(ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font>, <font color=red>(1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+min({{overline|sg}}(ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>120</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>, <font color=red>min((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>121</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font>, <font color=red>min((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x)),(1-μ<sub>B</sub>(x)).(μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(μ<sub>B</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>122</sub> | |||
| | |||
| | |||
| {<x, <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font>, <font color=red>((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x))).(1-μ<sub>B</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>123</sub> | |||
| | |||
| | |||
| {<x, <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x).(μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font>, <font color=red>((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x))).(((1-μ<sub>B</sub>(x)).(μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x))))+{{overline|sg}}(μ<sub>B</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>124</sub> | |||
| | |||
| | |||
| {<x, <font color=green>μ<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),ν<sub>A</sub>(x))</font>, <font color=red>ν<sub>B</sub>(x).μ<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>125</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>, <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>126</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font>, <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>127</sub> | |||
| | |||
| | |||
| {<x, <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x)-μ<sub>B</sub>(x).ν<sub>A</sub>(x)</font>, <font color=red>(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x))).μ<sub>A</sub>(x)</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>128</sub> | |||
| | |||
| | |||
| {<x, <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x).μ<sub>A</sub>(x)-μ<sub>B</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font>, <font color=red>(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x))).(μ<sub>A</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>129</sub> | |||
| | |||
| | |||
| {<x, <font color=green>1-ν<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font>, <font color=red>ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+min({{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>130</sub> | |||
| | |||
| | |||
| {<x, <font color=green>1-min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>, <font color=red>min(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>131</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(1-ν<sub>B</sub>(x),(1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))</font>, <font color=red>min(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>132</sub> | |||
| | |||
| | |||
| {<x, <font color=green>1-μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font>, <font color=red>(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x))).μ<sub>A</sub>(x)</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>133</sub> | |||
| | |||
| | |||
| {<x, <font color=green>1-ν<sub>B</sub>(x)+(1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)-(1-ν<sub>B</sub>(x)).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))</font>, <font color=red>(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x))).(μ<sub>A</sub>(x).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>134</sub> | |||
| | |||
| | |||
| {<x, <font color=green>μ<sub>B</sub>(x)+min({{overline|sg}}(μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font>, <font color=red>(1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+min({{overline|sg}}(μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>135</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>, <font color=red>min((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>136</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font>, <font color=red>min((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x)),(1-ν<sub>A</sub>(x)).(ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(ν<sub>A</sub>(x)))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>137</sub> | |||
| | |||
| | |||
| {<x, <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x)-μ<sub>B</sub>(x).ν<sub>A</sub>(x)</font>, <font color=red>((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x))).(1-ν<sub>A</sub>(x))</font>>|x ∈ E} | |||
|- valign="top" | |||
| →<sub>138</sub> | |||
| | |||
| | |||
| {<x, <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))-μ<sub>B</sub>(x).</font>, <font color=red></font>>|x ∈ E} | |||
|} | |||
--> | |||
=== List of implications === | |||
{| width="100%" class="wikitable" style="font-family:Courier; font-size:120%;" | |||
|- valign="top" | |||
! width="5%" | No. | |||
! width="5%" | Ref. | |||
! width="5%" | Year | |||
! width="85%" | Implication: | |||
{<x, <font color=green>Implication MEMBERSHIP expression</font>, <font color=red>Implication NON-MEMBERSHIP expression</font> >|x ∈ E} | |||
|} | |} | ||
{| width="100%" class="wikitable sortable" style="font-family:Courier; font-size:120%;" | |||
|- valign="top" | |||
! width="5%" | No. | |||
! width="5%" | Ref. | |||
! width="5%" | Year | |||
! width="40%" | Implication MEMBERSHIP expression<br/> | |||
! width="45%" | Implication NON-MEMBERSHIP expression<br/> | |||
|- valign="top" | |||
| →<sub>1</sub> | |||
| | |||
| | |||
| <font color=green> max(ν<sub>A</sub>(x),min(μ<sub>A</sub>(x),μ<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>2</sub> | |||
| | |||
| | |||
| <font color=green> {{overline|sg}}(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red> ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>3</sub> | |||
| | |||
| | |||
| <font color=green> 1-(1-μ<sub></sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>ν<sub>B</sub>.sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)) </font> | |||
|- valign="top" | |||
| →<sub>4</sub> | |||
| | |||
| | |||
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>5</sub> | |||
| | |||
| | |||
| <font color=green>min(1,ν<sub>A</sub>(x)+μ<sub>B</sub>(x))</font> || <font color=red>max(0,μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> | |||
|- valign="top" | |||
| →<sub>6</sub> | |||
| | |||
| | |||
| <font color=green>ν<sub>A</sub>(x)+μ<sub>A</sub>(x)μ<sub>B</sub>(x)</font> || <font color=red>μ<sub>A</sub>(x)ν<sub>B</sub>(x)</font> | |||
|- valign="top" | |||
| →<sub>7</sub> | |||
| | |||
| | |||
| <font color=green>min(max(ν<sub>A</sub>(x),μ<sub>B</sub>(x)),max(μ<sub>A</sub>(x),ν<sub>A</sub>(x)), max(μ<sub>B</sub>(x),ν<sub>B</sub>(x)))</font> || <font color=red>max(min(μ<sub>A</sub>(x),ν<sub>B</sub>(x)), min(μ<sub>A</sub>(x),ν<sub>A</sub>(x)),min(μ<sub>B</sub>(x),ν<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>8</sub> | |||
| | |||
| | |||
| <font color=green>1-(1-min(ν<sub>A</sub>(x),μ<sub>B</sub>(x))).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>max(μ<sub>A</sub>(x),ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)),sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>9</sub> | |||
| | |||
| | |||
| <font color=green>ν<sub>A</sub>(x)+μ<sub>A</sub>(x)<sup>2</sup>μ<sub>B</sub>(x)</font> || <font color=red>μ<sub>A</sub>(x)ν<sub>A</sub>(x)+μ<sub>A</sub>(x)<sup>2</sup>ν<sub>B</sub>(x)</font> | |||
|- valign="top" | |||
| →<sub>10</sub> | |||
| | |||
| | |||
| <font color=green>μ<sub>A</sub>(x).{{overline|sg}}(1-μ<sub>A</sub>(x))+sg(1-μ<sub>A</sub>(x)).({{overline|sg}}(1-μ<sub>B</sub>(x))+ν<sub>A</sub>(x).sg(1-μ<sub>B</sub>(x)))</font> || <font color=red>ν<sub>B</sub>.{{overline|sg}}(1-μ<sub>A</sub>(x))+μ<sub>A</sub>(x).sg(1-μ<sub>A</sub>(x)).sg(1-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>11</sub> | |||
| | |||
| | |||
| <font color=green>1-(1-μ<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>12</sub> | |||
| | |||
| | |||
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>13</sub> | |||
| | |||
| | |||
| <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font> || <font color=red>μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font> | |||
|- valign="top" | |||
| →<sub>14</sub> | |||
| | |||
| | |||
| <font color=green>1-(1-μ<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))-ν<sub>B</sub>(x).{{overline|sg}}(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>ν<sub>B</sub>(x).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>15</sub> | |||
| | |||
| | |||
| <font color=green>1-sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>sg({{overline|sg}}(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))+{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>16</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>17</sub> | |||
| | |||
| | |||
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+μ<sub>A</sub>(x)<sup>2</sup>,ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>18</sub> | |||
| | |||
| | |||
| <font color=green> max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>min(1-ν<sub>A</sub>(x),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>19</sub> | |||
| | |||
| | |||
| <font color=green>max(1-sg(sg(μ<sub>A</sub>(x))+sg(1-ν<sub>A</sub>(x))),μ<sub>B</sub>(x))</font> || <font color=red>min(sg(1-ν<sub>A</sub>(x)),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>20</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),sg(μ<sub>A</sub>(x))))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),{{overline|sg}}(μ<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>21</sub> | |||
| | |||
| | |||
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x).(μ<sub>B</sub>(x)+ν<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x).(μ<sub>A</sub>(x)+ν<sub>A</sub>(x)),ν<sub>B</sub>(x).(μ<sub>B</sub>(x)<sup>2</sup>+ν<sub>B</sub>(x)+μ<sub>B</sub>(x).ν<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>22</sub> | |||
| | |||
| | |||
| <font color=green>max(ν<sub>A</sub>(x),1-ν<sub>B</sub>(x))</font> || <font color=red>min(1-ν<sub>A</sub>(x),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>23</sub> | |||
| | |||
| | |||
| <font color=green>1-min(sg(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font> || <font color=red>min(sg(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>24</sub> | |||
| | |||
| | |||
| <font color=green>{{overline|sg}}(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>25</sub> | |||
| | |||
| | |||
| <font color=green>max(ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>A</sub>(x)).{{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x).{{overline|sg}}(ν<sub>B</sub>(x)).{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>26</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>27</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),sg(μ<sub>B</sub>(x)))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>28</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>29</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>30</sub> | |||
| | |||
| | |||
| <font color=green>max(1-μ<sub>A</sub>(x),min(μ<sub>A</sub>(x),1-ν<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>31</sub> | |||
| | |||
| | |||
| <font color=green>{{overline|sg}}(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> || <font color=red>ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> | |||
|- valign="top" | |||
| →<sub>32</sub> | |||
| | |||
| | |||
| <font color=green>1-ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> || <font color=red>ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> | |||
|- valign="top" | |||
| →<sub>33</sub> | |||
| | |||
| | |||
| <font color=green>1-min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>34</sub> | |||
| | |||
| | |||
| <font color=green>min(1,2-μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>max(0,μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> | |||
|- valign="top" | |||
| →<sub>35</sub> | |||
| | |||
| | |||
| <font color=green>1-μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font> || <font color=red>μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font> | |||
|- valign="top" | |||
| →<sub>36</sub> | |||
| | |||
| | |||
| <font color=green>min(1-min(μ<sub>A</sub>(x),ν<sub>B</sub>(x)),max(μ<sub>A</sub>(x),1-μ<sub>A</sub>(x)),max(1-ν<sub>B</sub>(x),ν<sub>B</sub>(x)))</font> || <font color=red>max(min(μ<sub>A</sub>(x),ν<sub>B</sub>(x)),min(μ<sub>A</sub>(x),1-μ<sub>A</sub>(x)),min(1-ν<sub>B</sub>(x),ν<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>37</sub> | |||
| | |||
| | |||
| <font color=green>1-max(μ<sub>A</sub>(x),ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> || <font color=red>max(μ<sub>A</sub>(x),ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> | |||
|- valign="top" | |||
| →<sub>38</sub> | |||
| | |||
| | |||
| <font color=green>1-μ<sub>A</sub>(x)+(μ<sub>A</sub>(x)<sup>2</sup>.(1-ν<sub>B</sub>(x)))</font> || <font color=red>μ<sub>A</sub>(x)(1-μ<sub>A</sub>(x))+μ<sub>A</sub>(x)<sup>2</sup>.ν<sub>B</sub>(x)</font> | |||
|- valign="top" | |||
| →<sub>39</sub> | |||
| | |||
| | |||
| <font color=green>(1-ν<sub>B</sub>(x)).{{overline|sg}}(1-μ<sub>A</sub>(x))+sg(1-μ<sub>A</sub>(x)).({{overline|sg}}(ν<sub>B</sub>(x))+(1-μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)))</font> || <font color=red>ν<sub>B</sub>(x).{{overline|sg}}(1-μ<sub>A</sub>(x))+μ<sub>A</sub>(x).sg(1-μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>40</sub> | |||
| | |||
| | |||
| <font color=green>1-sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> || <font color=red>1-{{overline|sg}}(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> | |||
|- valign="top" | |||
| →<sub>41</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>42</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),sg(1-ν<sub>B</sub>(x)))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>43</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>44</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font> | |||
=== | |- valign="top" | ||
| →<sub>45</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),{{overline|sg}}(ν<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>46</sub> | |||
| | |||
| | |||
| <font color=green>max(ν<sub>A</sub>(x),min(1-ν<sub>A</sub>(x),μ<sub>B</sub>(x)))</font> || <font color=red>1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>47</sub> | |||
| | |||
| | |||
| <font color=green>{{overline|sg}}(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>(1-μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>48</sub> | |||
| | |||
| | |||
| <font color=green>1-(1-μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>(1-μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>49</sub> | |||
| | |||
| | |||
| <font color=green>min(1,ν<sub>A</sub>(x)+μ<sub>B</sub>(x))</font> || <font color=red>max(0,1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>50</sub> | |||
| | |||
| | |||
| <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font> || <font color=red>1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x)+ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font> | |||
|- valign="top" | |||
| →<sub>51</sub> | |||
| | |||
| | |||
| <font color=green>min(max(ν<sub>A</sub>(x),μ<sub>B</sub>(x)),max(1-ν<sub>A</sub>(x),ν<sub>A</sub>(x)),max(μ<sub>B</sub>(x),1-μ<sub>B</sub>(x)))</font> || <font color=red>max(1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x)),min(1-ν<sub>A</sub>(x),ν<sub>A</sub>(x)),min(μ<sub>B</sub>(x),1-μ<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>52</sub> | |||
| | |||
| | |||
| <font color=green>1-(1-min(ν<sub>A</sub>(x),μ<sub>B</sub>(x))).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>1-min(ν<sub>A</sub>(x),μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>53</sub> | |||
| | |||
| | |||
| <font color=green>ν<sub>A</sub>(x)+(1-ν<sub>A</sub>(x))<sup>2</sup>.μ<sub>B</sub>(x)</font> || <font color=red>(1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+(1-ν<sub>A</sub>(x))<sup>2</sup>.(1-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>54</sub> | |||
| | |||
| | |||
| <font color=green>μ<sub>B</sub>(x){{overline|sg}}(ν<sub>A</sub>(x))+sg(ν<sub>A</sub>(x)).({{overline|sg}}(1-μ<sub>B</sub>(x))+ν<sub>A</sub>(x).sg(1-μ<sub>B</sub>(x)))</font> || <font color=red>(1-μ<sub>B</sub>(x)).{{overline|sg}}(ν<sub>A</sub>(x))+(1-ν<sub>A</sub>(x)).sg(ν<sub>A</sub>(x)).sg(1-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>55</sub> | |||
| | |||
| | |||
| <font color=green>1-sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>1-{{overline|sg}}(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>56</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>min(sg(1-ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>57</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),sg(μ<sub>B</sub>(x)))</font> || <font color=red>min(sg(1-ν<sub>A</sub>(x)),{{overline|sg}}(μ<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>58</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>59</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>60</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>min(1-ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>61</sub> | |||
| | |||
| | |||
| <font color=green>max(μ<sub>B</sub>(x),min(ν<sub>B</sub>(x),ν<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>62</sub> | |||
| | |||
| | |||
| <font color=green>{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>63</sub> | |||
| | |||
| | |||
| <font color=green>1-(1-ν<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>64</sub> | |||
| | |||
| | |||
| <font color=green>μ<sub>B</sub>(x)+ν<sub>B</sub>(x).ν<sub>A</sub>(x)</font> || <font color=red>ν<sub>B</sub>(x).μ<sub>A</sub>(x)</font> | |||
|- valign="top" | |||
| →<sub>65</sub> | |||
| | |||
| | |||
| <font color=green>1-(1-min(μ<sub>B</sub>(x),ν<sub>A</sub>(x))).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>max(ν<sub>B</sub>(x),μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>66</sub> | |||
| | |||
| | |||
| <font color=green>μ<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>ν<sub>A</sub>(x)</font> || <font color=red>ν<sub>B</sub>(x).μ<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>μ<sub>A</sub>(x)</font> | |||
|- valign="top" | |||
| →<sub>67</sub> | |||
| | |||
| | |||
| <font color=green>ν<sub>A</sub>(x).{{overline|sg}}(1-ν<sub>B</sub>(x))+sg(1-ν<sub>B</sub>(x)).({{overline|sg}}(1-ν<sub>A</sub>(x))+μ<sub>B</sub>(x).sg(1-ν<sub>A</sub>(x)))</font> || <font color=red>μ<sub>A</sub>(x).{{overline|sg}}(1-ν<sub>B</sub>(x))+ν<sub>B</sub>(x).sg(1-ν<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>68</sub> | |||
| | |||
| | |||
| <font color=green>1-(1-ν<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>69</sub> | |||
| | |||
| | |||
| <font color=green>1-(1-ν<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))-μ<sub>A</sub>(x).{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>70</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}((ν<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>min(sg(ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>71</sub> | |||
| | |||
| | |||
| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font> || <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>,μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>72</sub> | |||
| | |||
| | |||
| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font> || <font color=red>min(1-μ<sub>B</sub>(x),μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>73</sub> | |||
| | |||
| | |||
| <font color=green>max(1-max(sg(ν<sub>B</sub>(x)),sg(1-μ<sub>B</sub>(x))),ν<sub>A</sub>(x))</font> || <font color=red>min(sg(1-μ<sub>B</sub>(x)),μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>74</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),sg(ν<sub>A</sub>(x)))</font> || <font color=red>min(sg(ν<sub>B</sub>(x)),{{overline|sg}}(ν<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>75</sub> | |||
| | |||
| | |||
| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).(ν<sub>A</sub>(x)+μ<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x).(ν<sub>B</sub>(x)+μ<sub>B</sub>(x)),μ<sub>A</sub>(x).(ν<sub>A</sub>(x)<sup>2</sup>+μ<sub>A</sub>(x))+ν<sub>A</sub>(x).μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>76</sub> | |||
| | |||
| | |||
| <font color=green>max(μ<sub>B</sub>(x),1-μ<sub>A</sub>(x))</font> || <font color=red>min(1-μ<sub>B</sub>(x),μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>77</sub> | |||
| | |||
| | |||
| <font color=green>1-min(sg(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font> || <font color=red>min(sg(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>78</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>min(sg(ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>80</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>81</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-ν<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>82</sub> | |||
| | |||
| | |||
| <font color=green>max(1-ν<sub>B</sub>(x),min(ν<sub>B</sub>(x),1-μ<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>83</sub> | |||
| | |||
| | |||
| <font color=green>{{overline|sg}}(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)-1)</font> || <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)-1)</font> | |||
|- valign="top" | |||
| →<sub>84</sub> | |||
| | |||
| | |||
| <font color=green>1-μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)+1)</font> || <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)+1)</font> | |||
|- valign="top" | |||
| →<sub>85</sub> | |||
| | |||
| | |||
| <font color=green>1-ν<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>.(1-μ<sub>A</sub>(x))</font> || <font color=red>ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+ν<sub>B</sub>(x)<sup>2</sup></font> | |||
|- valign="top" | |||
| →<sub>86</sub> | |||
| | |||
| | |||
| <font color=green>(1-μ<sub>A</sub>(x)).{{overline|sg}}(1-ν<sub>B</sub>(x))+sg(1-ν<sub>B</sub>(x)){{overline|sg}}(μ<sub>A</sub>(x)+min(1-ν<sub>B</sub>(x),sg(μ<sub>A</sub>(x))))</font> || <font color=red>μ<sub>A</sub>(x).{{overline|sg}}(1-ν<sub>B</sub>(x))+ν<sub>B</sub>(x).sg(1-ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>87</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font> || <font color=red>min(sg(ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>88</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),sg(1-μ<sub>A</sub>(x)))</font> || <font color=red>min(sg(ν<sub>B</sub>(x)),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>89</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font> || <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>90</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),{{overline|sg}}(μ<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>91</sub> | |||
| | |||
| | |||
| <font color=green>max(μ<sub>B</sub>(x),min(1-μ<sub>B</sub>(x),ν<sub>A</sub>(x)))</font> || <font color=red>1-max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>92</sub> | |||
| | |||
| | |||
| <font color=green>{{overline|sg}}(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>93</sub> | |||
| | |||
| | |||
| <font color=green>1-min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font> || <font color=red>min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>94</sub> | |||
| | |||
| | |||
| <font color=green>μ<sub>B</sub>(x)+(1-μ<sub>B</sub>(x))<sup>2</sup>.ν<sub>A</sub>(x))</font> || <font color=red>(1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+(1-μ<sub>B</sub>(x))<sup>2</sup>.(1-ν<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>95</sub> | |||
| | |||
| | |||
| <font color=green>min(ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))+sg(μ<sub>B</sub>(x)).({{overline|sg}}(1-ν<sub>A</sub>(x))+min(μ<sub>B</sub>(x),sg(1-ν<sub>A</sub>(x))))</font> || <font color=red>min(1-ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))+min(min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))),sg(1-ν<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>96</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>min(sg(1-μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x)</font> | |||
|- valign="top" | |||
| →<sub>97</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),sg(ν<sub>A</sub>(x)))</font> || <font color=red>min(sg(1-μ<sub>B</sub>(x)),{{overline|sg}}(ν<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>98</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>1-max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>99</sub> | |||
| | |||
| | |||
| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-ν<sub>A</sub>(x)))</font> || <font color=red>min(1-μ<sub>B</sub>(x),{{overline|sg}}(ν<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>100</sub> | |||
| | |||
| | |||
| <font color=green>max(min(ν<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),μ<sub>B</sub>(x))</font> || <font color=red>min(min(μ<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>101</sub> | |||
| | |||
| | |||
| <font color=green>max(min(ν<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),min(μ<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font> || <font color=red>min(min(μ<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),min(ν<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font> | |||
|- valign="top" | |||
| →<sub>102</sub> | |||
| | |||
| | |||
| <font color=green>max(ν<sub>A</sub>(x),min(μ<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font> || <font color=red>min(μ<sub>A</sub>(x),min(ν<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font> | |||
|- valign="top" | |||
| →<sub>103</sub> | |||
| | |||
| | |||
| <font color=green>max(min(1-μ<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),1-ν<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x),sg(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>104</sub> | |||
| | |||
| | |||
| <font color=green>max(min(1-μ<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),min(1-ν<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font> || <font color=red>min(min(μ<sub>A</sub>(x),sg(1-μ<sub>A</sub>(x))),min(ν<sub>B</sub>(x),sg(1-ν<sub>B</sub>(x))))</font> | |||
|- valign="top" | |||
| →<sub>105</sub> | |||
| | |||
| | |||
| <font color=green>max(1-μ<sub>A</sub>(x),min(1-ν<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font> || <font color=red>min(μ<sub>A</sub>(x),min(ν<sub>B</sub>(x),sg(1-ν<sub>B</sub>(x))))</font> | |||
|- valign="top" | |||
| →<sub>106</sub> | |||
| | |||
| | |||
| <font color=green>max(min(ν<sub>A</sub>(x),sg(1-ν<sub>A</sub>(x))),μ<sub>B</sub>(x))</font> || <font color=red>min(min(1-ν<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),1-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>107</sub> | |||
| | |||
| | |||
| <font color=green>max(min(ν<sub>A</sub>(x),sg(1-ν<sub>A</sub>(x))),min(μ<sub>B</sub>(x),sg(1-μ<sub>B</sub>(x))))</font> || <font color=red>min(min(1-ν<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font> | |||
|- valign="top" | |||
| →<sub>108</sub> | |||
| | |||
| | |||
| <font color=green>max(ν<sub>A</sub>(x),min(μ<sub>B</sub>(x),sg(1-μ<sub>B</sub>(x))))</font> || <font color=red>min(1-ν<sub>A</sub>(x),min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font> | |||
|- valign="top" | |||
| →<sub>109</sub> | |||
| | |||
| | |||
| <font color=green>ν<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).ν<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>110</sub> | |||
| | |||
| | |||
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>111</sub> | |||
| | |||
| | |||
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>112</sub> | |||
| | |||
| | |||
| <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font> || <font color=red>μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)).ν<sub>B</sub>(x)</font> | |||
|- valign="top" | |||
| →<sub>113</sub> | |||
| | |||
| | |||
| <font color=green>ν<sub>A</sub>(x)+(μ<sub>B</sub>(x).ν<sub>B</sub>(x)-ν<sub>A</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x))).(ν<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>114</sub> | |||
| | |||
| | |||
| <font color=green>1-μ<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+min({{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>115</sub> | |||
| | |||
| | |||
| <font color=green>1-min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x)(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>116</sub> | |||
| | |||
| | |||
| <font color=green>max(1-μ<sub>A</sub>(x),(1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)+{{overline|sg}}(ν<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x).((1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)+{{overline|sg}}(ν<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>117</sub> | |||
| | |||
| | |||
| <font color=green>1-μ<sub>A</sub>(x)-ν<sub>B</sub>(x)+μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font> || <font color=red>(μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x))).ν<sub>B</sub>(x)</font> | |||
|- valign="top" | |||
| →<sub>118</sub> | |||
| | |||
| | |||
| <font color=green>(1-μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x))+μ<sub>A</sub>(x).ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))</font> || <font color=red>(μ<sub>A</sub>(x)-μ<sub>A</sub>(x)<sup>2</sup>+{{overline|sg}}(1-μ<sub>A</sub>(x))).((1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)<sup>2</sup>+{{overline|sg}}(1-ν<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x))</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>119</sub> | |||
| | |||
| | |||
| <font color=green>ν<sub>A</sub>(x)+min({{overline|sg}}(ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>(1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+min({{overline|sg}}(ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>120</sub> | |||
| | |||
| | |||
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>min((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>121</sub> | |||
| | |||
| | |||
| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>min((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x)),(1-μ<sub>B</sub>(x)).(μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(μ<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>122</sub> | |||
| | |||
| | |||
| <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font> || <font color=red>((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x))).(1-μ<sub>B</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>123</sub> | |||
| | |||
| | |||
| <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x).(μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x))).(((1-μ<sub>B</sub>(x)).(μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x))))+{{overline|sg}}(μ<sub>B</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>124</sub> | |||
| | |||
| | |||
| <font color=green>μ<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>ν<sub>B</sub>(x).μ<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>125</sub> | |||
| | |||
| | |||
| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font> || <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>126</sub> | |||
| | |||
| | |||
| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>127</sub> | |||
| | |||
| | |||
| <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x)-μ<sub>B</sub>(x).ν<sub>A</sub>(x)</font> || <font color=red>(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x))).μ<sub>A</sub>(x)</font> | |||
|- valign="top" | |||
| →<sub>128</sub> | |||
| | |||
| | |||
| <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x).μ<sub>A</sub>(x)-μ<sub>B</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font> || <font color=red>(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x))).(μ<sub>A</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>129</sub> | |||
| | |||
| | |||
| <font color=green>1-ν<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font> || <font color=red>ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+min({{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>130</sub> | |||
| | |||
| | |||
| <font color=green>1-min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font> || <font color=red>min(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>131</sub> | |||
| | |||
| | |||
| <font color=green>max(1-ν<sub>B</sub>(x),(1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>132</sub> | |||
| | |||
| | |||
| <font color=green>1-μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font> || <font color=red>(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x))).μ<sub>A</sub>(x)</font> | |||
|- valign="top" | |||
| →<sub>133</sub> | |||
| | |||
| | |||
| <font color=green>1-ν<sub>B</sub>(x)+(1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)-(1-ν<sub>B</sub>(x)).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))</font> || <font color=red>(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x))).(μ<sub>A</sub>(x).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>134</sub> | |||
| | |||
| | |||
| <font color=green>μ<sub>B</sub>(x)+min({{overline|sg}}(μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>(1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+min({{overline|sg}}(μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>135</sub> | |||
| | |||
| | |||
| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font> || <font color=red>min((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>136</sub> | |||
| | |||
| | |||
| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font> || <font color=red>min((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x)),(1-ν<sub>A</sub>(x)).(ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(ν<sub>A</sub>(x)))</font> | |||
|- valign="top" | |||
| →<sub>137</sub> | |||
| | |||
| | |||
| <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x)-μ<sub>B</sub>(x).ν<sub>A</sub>(x)</font> || <font color=red>((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x))).(1-ν<sub>A</sub>(x))</font> | |||
|- valign="top" | |||
| →<sub>138</sub> | |||
| | |||
| | |||
| <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))-μ<sub>B</sub>(x).</font> || <font color=red> | |||
((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x))).(1-ν<sub>A</sub>(x).(ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x)) + {{overline|sg}}(1-ν<sub>A</sub>(x)) + {{overline|sg}}(ν<sub>A</sub>(x)) | |||
|- valign="top" | |||
| →<sub>139</sub> | |||
| | |||
| | |||
| <font color=green>(ν<sub>A</sub>(x) + μ<sub>B</sub>(x))/2</font> || <font color=red>(μ<sub>A</sub>(x) + ν<sub>B</sub>(x))/2</font> | |||
|- valign="top" | |||
| →<sub>140</sub> | |||
| | |||
| | |||
| <font color=green>(ν<sub>A</sub>(x) + μ<sub>B</sub>(x) + min(ν<sub>A</sub>(x), μ<sub>B</sub>(x)))/3</font> || <font color=red>(μ<sub>A</sub>(x) + ν<sub>B</sub>(x) + max(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font> | |||
|- valign="top" | |||
| →<sub>141</sub> | |||
| | |||
| | |||
| <font color=green>(ν<sub>A</sub>(x) + μ<sub>B</sub>(x) + max(ν<sub>A</sub>(x), μ<sub>B</sub>(x)))/3</font> || <font color=red>(μ<sub>A</sub>(x) + ν<sub>B</sub>(x) + min(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font> | |||
|- valign="top" | |||
| →<sub>142</sub> | |||
| | |||
| | |||
| <font color=green>(3 - μ<sub>A</sub>(x) - ν<sub>B</sub>(x) - max(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font> || <font color=red>(μ<sub>A</sub>(x) + ν<sub>B</sub>(x) + max(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font> | |||
|- valign="top" | |||
| →<sub>143</sub> | |||
| | |||
| | |||
| <font color=green>(1 - μ<sub>A</sub>(x) + μ<sub>b</sub>(x) + min(1 - μ<sub>A</sub>(x), μ<sub>B</sub>(x)) )/3</font> || <font color=red>(2 + μ<sub>A</sub>(x) - μ<sub>B</sub>(x) - min(1 - μ<sub>A</sub>(x), μ<sub>B</sub>(x)))/3</font> | |||
|- valign="top" | |||
| →<sub>144</sub> | |||
| | |||
| | |||
| <font color=green>(1 + ν<sub>A</sub>(x) - ν<sub>b</sub>(x) + min(ν<sub>A</sub>(x), 1 - ν<sub>B</sub>(x)) )/3</font> || <font color=red>(2 - ν<sub>A</sub>(x) + ν<sub>B</sub>(x) + min(ν<sub>A</sub>(x), 1 - ν<sub>B</sub>(x)))/3</font> | |||
|- valign="top" | |||
| →<sub>145</sub> | |||
| | |||
| | |||
| <font color=green>(ν<sub>A</sub>(x) + μ<sub>B</sub>(x) + min(ν<sub>A</sub>(x), μ<sub>B</sub>(x)))/3</font> || <font color=red>(3 - ν<sub>A</sub>(x) - μ<sub>B</sub>(x) - min(ν<sub>A</sub>(x), μ<sub>B</sub>(x)))/3</font> | |||
|- valign="top" | |||
| →<sub>146</sub> | |||
| | |||
| | |||
| <font color=green>(3 - μ<sub>A</sub>(x) - ν<sub>B</sub>(x) - min(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font> || <font color=red>(μ<sub>A</sub>(x) + ν<sub>B</sub>(x) + min(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font> | |||
|- valign="top" | |||
| →<sub>147</sub> | |||
| | |||
| | |||
| <font color=green>(1 - μ<sub>A</sub>(x) + μ<sub>b</sub>(x) + max(1 - μ<sub>A</sub>(x), μ<sub>B</sub>(x)) )/3</font> || <font color=red>(2 + μ<sub>A</sub>(x) - μ<sub>b</sub>(x) - max(1 - μ<sub>A</sub>(x), μ<sub>B</sub>(x)) )/3</font> | |||
</font> | |||
|} | |||
== References == | == References == | ||
| Line 548: | Line 1,759: | ||
* [[Issue:On some two-parametric intuitionistic fuzzy implications|On some two-parametric intuitionistic fuzzy implications]], Piotr Dworniczak, 2011 | * [[Issue:On some two-parametric intuitionistic fuzzy implications|On some two-parametric intuitionistic fuzzy implications]], Piotr Dworniczak, 2011 | ||
* [[Issue:Second Zadeh's intuitionistic fuzzy implication|Second Zadeh's intuitionistic fuzzy implication]], Krassimir Atanassov, 2011 | * [[Issue:Second Zadeh's intuitionistic fuzzy implication|Second Zadeh's intuitionistic fuzzy implication]], Krassimir Atanassov, 2011 | ||
; "What Links Here" References | |||
{{Special:WhatLinksHere/{{PAGENAME}}|namespace=102|hidetrans=1|hideredirs=1}} | |||
== See also == | == See also == | ||
Latest revision as of 09:41, 29 April 2022
For the various definitions of implication of over intuitionistic fuzzy sets, the functions sg(x) and sg(x) have been used:
[math]\displaystyle{ \text{sg}(x) = \begin{cases} 1 \text{ if } x \gt 0 \\ 0 \text{ if } x \leq 0 \end{cases}, }[/math] [math]\displaystyle{ \overline{\text{sg}}(x) = \begin{cases} 0 \text{ if } x \gt 0 \\ 1 \text{ if } x \leq 0 \end{cases}. }[/math]
List of implications
| No. | Ref. | Year | Implication:
{<x, Implication MEMBERSHIP expression, Implication NON-MEMBERSHIP expression >|x ∈ E} |
|---|
| No. | Ref. | Year | Implication MEMBERSHIP expression |
Implication NON-MEMBERSHIP expression |
|---|---|---|---|---|
| →1 | max(νA(x),min(μA(x),μB(x))) | min(μA(x),νB(x)) | ||
| →2 | sg(μA(x)-μB(x)) | νB(x).sg(μA(x)-μB(x)) | ||
| →3 | 1-(1-μ(x)).sg(μA(x)-μB(x)) | νB.sg(μA(x)-μB(x)) | ||
| →4 | max(νA(x),μB(x)) | min(μA(x),νB(x)) | ||
| →5 | min(1,νA(x)+μB(x)) | max(0,μA(x)+νB(x)-1) | ||
| →6 | νA(x)+μA(x)μB(x) | μA(x)νB(x) | ||
| →7 | min(max(νA(x),μB(x)),max(μA(x),νA(x)), max(μB(x),νB(x))) | max(min(μA(x),νB(x)), min(μA(x),νA(x)),min(μB(x),νB(x))) | ||
| →8 | 1-(1-min(νA(x),μB(x))).sg(μA(x)-μB(x)) | max(μA(x),νB(x)).sg(μA(x)-μB(x)),sg(νB(x)-νA(x)) | ||
| →9 | νA(x)+μA(x)2μB(x) | μA(x)νA(x)+μA(x)2νB(x) | ||
| →10 | μA(x).sg(1-μA(x))+sg(1-μA(x)).(sg(1-μB(x))+νA(x).sg(1-μB(x))) | νB.sg(1-μA(x))+μA(x).sg(1-μA(x)).sg(1-μB(x)) | ||
| →11 | 1-(1-μB(x)).sg(μA(x)-μB(x)) | νB(x).sg(μA(x)-μB(x)).sg(νB(x)-νA(x)) | ||
| →12 | max(νA(x),μB(x)) | 1-max(νA(x),μB(x)) | ||
| →13 | νA(x)+μB(x)-νA(x).μB(x) | μA(x).νB(x) | ||
| →14 | 1-(1-μB(x)).sg(μA(x)-μB(x))-νB(x).sg(μA(x)-μB(x)).sg(νB(x)-νA(x)) | νB(x).sg(νB(x)-νA(x)) | ||
| →15 | 1-sg(μA(x)-μB(x)).sg(νB(x)-νA(x)) | sg(sg(μA(x)-μB(x))+sg(νB(x)-νA(x))) | ||
| →16 | max(sg(μA(x)),μB(x)) | min(sg(μA(x)),νB(x)) | ||
| →17 | max(νA(x),μB(x)) | min(μA(x).νA(x)+μA(x)2,νB(x)) | ||
| →18 | max(νA(x),μB(x)) | min(1-νA(x),νB(x)) | ||
| →19 | max(1-sg(sg(μA(x))+sg(1-νA(x))),μB(x)) | min(sg(1-νA(x)),νB(x)) | ||
| →20 | max(sg(μA(x)),sg(μA(x)))) | min(sg(μA(x)),sg(μB(x))) | ||
| →21 | max(νA(x),μB(x).(μB(x)+νB(x))) | min(μA(x).(μA(x)+νA(x)),νB(x).(μB(x)2+νB(x)+μB(x).νB(x))) | ||
| →22 | max(νA(x),1-νB(x)) | min(1-νA(x),νB(x)) | ||
| →23 | 1-min(sg(1-νA(x)),sg(1-νB(x))) | min(sg(1-νA(x)),sg(1-νB(x))) | ||
| →24 | sg(μA(x)-μB(x)).sg(νB(x)-νA(x)) | sg(μA(x)-μB(x)).sg(νB(x)-νA(x)) | ||
| →25 | max(νA(x),sg(μA(x)).sg(1-νA(x)),μB(x).sg(νB(x)).sg(1-μB(x))) | min(μA(x),νB(x)) | ||
| →26 | max(sg(1-νA(x)),μB(x)) | min(sg(μA(x)),νB(x)) | ||
| →27 | max(sg(1-νA(x)),sg(μB(x))) | min(sg(μA(x)),sg(1-νB(x))) | ||
| →28 | max(sg(1-νA(x)),μB(x)) | min(μA(x),νB(x)) | ||
| →29 | max(sg(1-νA(x)),sg(1-μB(x))) | min(μA(x),sg(1-νB(x))) | ||
| →30 | max(1-μA(x),min(μA(x),1-νB(x))) | min(μA(x),νB(x)) | ||
| →31 | sg(μA(x)+νB(x)-1) | νB(x).sg(μA(x)+νB(x)-1) | ||
| →32 | 1-νB(x).sg(μA(x)+νB(x)-1) | νB(x).sg(μA(x)+νB(x)-1) | ||
| →33 | 1-min(μA(x),νB(x)) | min(μA(x),νB(x)) | ||
| →34 | min(1,2-μA(x)-μB(x)) | max(0,μA(x)+νB(x)-1) | ||
| →35 | 1-μA(x).νB(x) | μA(x).νB(x) | ||
| →36 | min(1-min(μA(x),νB(x)),max(μA(x),1-μA(x)),max(1-νB(x),νB(x))) | max(min(μA(x),νB(x)),min(μA(x),1-μA(x)),min(1-νB(x),νB(x))) | ||
| →37 | 1-max(μA(x),νB(x)).sg(μA(x)+νB(x)-1) | max(μA(x),νB(x)).sg(μA(x)+νB(x)-1) | ||
| →38 | 1-μA(x)+(μA(x)2.(1-νB(x))) | μA(x)(1-μA(x))+μA(x)2.νB(x) | ||
| →39 | (1-νB(x)).sg(1-μA(x))+sg(1-μA(x)).(sg(νB(x))+(1-μA(x)).sg(νB(x))) | νB(x).sg(1-μA(x))+μA(x).sg(1-μA(x)).sg(νB(x)) | ||
| →40 | 1-sg(μA(x)+νB(x)-1) | 1-sg(μA(x)+νB(x)-1) | ||
| →41 | max(sg(μA(x)),1-νB(x)) | min(sg(μA(x)),νB(x)) | ||
| →42 | max(sg(μA(x)),sg(1-νB(x))) | min(sg(μA(x)),sg(1-νB(x))) | ||
| →43 | max(sg(μA(x)),1-νB(x)) | min(sg(μA(x)),νB(x)) | ||
| →44 | max(sg(μA(x)),1-νB(x)) | min(μA(x),νB(x)) | ||
| →45 | max(sg(μA(x)),sg(νB(x))) | min(μA(x),sg(1-νB(x))) | ||
| →46 | max(νA(x),min(1-νA(x),μB(x))) | 1-max(νA(x),μB(x)) | ||
| →47 | sg(1-νA(x)-μB(x)) | (1-μB(x)).sg(1-νA(x)-μB(x)) | ||
| →48 | 1-(1-μB(x)).sg(1-νA(x)-μB(x)) | (1-μB(x)).sg(1-νA(x)-μB(x)) | ||
| →49 | min(1,νA(x)+μB(x)) | max(0,1-νA(x)-μB(x)) | ||
| →50 | νA(x)+μB(x)-νA(x).μB(x) | 1-νA(x)-μB(x)+νA(x).μB(x) | ||
| →51 | min(max(νA(x),μB(x)),max(1-νA(x),νA(x)),max(μB(x),1-μB(x))) | max(1-max(νA(x),μB(x)),min(1-νA(x),νA(x)),min(μB(x),1-μB(x))) | ||
| →52 | 1-(1-min(νA(x),μB(x))).sg(1-νA(x)-μB(x)) | 1-min(νA(x),μB(x)).sg(1-νA(x)-μB(x)) | ||
| →53 | νA(x)+(1-νA(x))2.μB(x) | (1-νA(x)).νA(x)+(1-νA(x))2.(1-μB(x)) | ||
| →54 | μB(x)sg(νA(x))+sg(νA(x)).(sg(1-μB(x))+νA(x).sg(1-μB(x))) | (1-μB(x)).sg(νA(x))+(1-νA(x)).sg(νA(x)).sg(1-μB(x)) | ||
| →55 | 1-sg(1-νA(x)-μB(x)) | 1-sg(1-νA(x)-μB(x)) | ||
| →56 | max(sg(1-νA(x)),μB(x)) | min(sg(1-νA(x)),1-μB(x)) | ||
| →57 | max(sg(1-νA(x)),sg(μB(x))) | min(sg(1-νA(x)),sg(μB(x))) | ||
| →58 | max(sg(1-νA(x)),sg(1-μB(x))) | 1-max(νA(x),μB(x)) | ||
| →59 | max(sg(1-νA(x)),μB(x)) | 1-max(νA(x),μB(x)) | ||
| →60 | max(sg(1-νA(x)),sg(1-μB(x))) | min(1-νA(x),sg(μB(x))) | ||
| →61 | max(μB(x),min(νB(x),νA(x))) | min(νB(x),μA(x)) | ||
| →62 | sg(νB(x)-νA(x)) | μA(x).sg(νB(x)-νA(x)) | ||
| →63 | 1-(1-νA(x)).sg(νB(x)-νA(x)) | μA(x).sg(νB(x)-νA(x)) | ||
| →64 | μB(x)+νB(x).νA(x) | νB(x).μA(x) | ||
| →65 | 1-(1-min(μB(x),νA(x))).sg(νB(x)-νA(x)) | max(νB(x),μA(x)).sg(νB(x)-νA(x)).sg(μA(x)-μB(x)) | ||
| →66 | μB(x)+νB(x)2νA(x) | νB(x).μB(x)+νB(x)2μA(x) | ||
| →67 | νA(x).sg(1-νB(x))+sg(1-νB(x)).(sg(1-νA(x))+μB(x).sg(1-νA(x))) | μA(x).sg(1-νB(x))+νB(x).sg(1-νB(x)).sg(1-νA(x)) | ||
| →68 | 1-(1-νA(x)).sg(νB(x)-νA(x)) | μA(x).sg(νB(x)-νA(x)).sg(μA(x)-μB(x)) | ||
| →69 | 1-(1-νA(x)).sg(νB(x)-νA(x))-μA(x).sg(νB(x)-νA(x)).sg(μA(x)-μB(x)) | μA(x).sg(μA(x)-μB(x)) | ||
| →70 | max(sg((νB(x)),νA(x)) | min(sg(νB(x)),μA(x)) | ||
| →71 | max(μB(x),νA(x)) | min(νB(x).μB(x)+νB(x)2,μA(x)) | ||
| →72 | max(μB(x),νA(x)) | min(1-μB(x),μA(x)) | ||
| →73 | max(1-max(sg(νB(x)),sg(1-μB(x))),νA(x)) | min(sg(1-μB(x)),μA(x)) | ||
| →74 | max(sg(νB(x)),sg(νA(x))) | min(sg(νB(x)),sg(νA(x))) | ||
| →75 | max(μB(x),νA(x).(νA(x)+μA(x))) | min(νB(x).(νB(x)+μB(x)),μA(x).(νA(x)2+μA(x))+νA(x).μA(x)) | ||
| →76 | max(μB(x),1-μA(x)) | min(1-μB(x),μA(x)) | ||
| →77 | 1-min(sg(1-μB(x)),sg(1-μA(x))) | min(sg(1-μB(x)),sg(1-μA(x))) | ||
| →78 | max(sg(1-μB(x)),νA(x)) | min(sg(νB(x)),μA(x)) | ||
| →80 | max(sg(1-μB(x)),νA(x)) | min(νB(x),μA(x)) | ||
| →81 | max(sg(1-μB(x)),sg(1-νA(x))) | min(νB(x),sg(1-μA(x))) | ||
| →82 | max(1-νB(x),min(νB(x),1-μA(x))) | min(νB(x),μA(x)) | ||
| →83 | sg(νB(x)+μA(x)-1) | μA(x).sg(νB(x)+μA(x)-1) | ||
| →84 | 1-μA(x).sg(νB(x)+μA(x)+1) | μA(x).sg(νB(x)+μA(x)+1) | ||
| →85 | 1-νB(x)+νB(x)2.(1-μA(x)) | νB(x).(1-νB(x))+νB(x)2 | ||
| →86 | (1-μA(x)).sg(1-νB(x))+sg(1-νB(x))sg(μA(x)+min(1-νB(x),sg(μA(x)))) | μA(x).sg(1-νB(x))+νB(x).sg(1-νB(x)).sg(μA(x)) | ||
| →87 | max(sg(νB(x)),1-μA(x)) | min(sg(νB(x)),μA(x)) | ||
| →88 | max(sg(νB(x)),sg(1-μA(x))) | min(sg(νB(x)),sg(1-μA(x))) | ||
| →89 | max(sg(νB(x)),1-μA(x)) | min(νB(x),μA(x)) | ||
| →90 | max(sg(νB(x)),sg(μA(x))) | min(νB(x),sg(1-μA(x))) | ||
| →91 | max(μB(x),min(1-μB(x),νA(x))) | 1-max(μB(x),νA(x)) | ||
| →92 | sg(1-μB(x)-νA(x)) | min(1-νA(x),sg(1-μB(x)-νA(x))) | ||
| →93 | 1-min(1-νA(x),sg(1-μB(x)-νA(x))) | min(1-νA(x),sg(1-μB(x)-νA(x))) | ||
| →94 | μB(x)+(1-μB(x))2.νA(x)) | (1-μB(x)).μB(x)+(1-μB(x))2.(1-νA(x)) | ||
| →95 | min(νA(x),sg(μB(x)))+sg(μB(x)).(sg(1-νA(x))+min(μB(x),sg(1-νA(x)))) | min(1-νA(x),sg(μB(x)))+min(min(1-μB(x),sg(μB(x))),sg(1-νA(x))) | ||
| →96 | max(sg(1-μB(x)),νA(x)) | min(sg(1-μB(x)),1-νA(x) | ||
| →97 | max(sg(1-μB(x)),sg(νA(x))) | min(sg(1-μB(x)),sg(νA(x))) | ||
| →98 | max(sg(1-μB(x)),νA(x)) | 1-max(μB(x),νA(x)) | ||
| →99 | max(sg(1-μB(x)),sg(1-νA(x))) | min(1-μB(x),sg(νA(x))) | ||
| →100 | max(min(νA(x),sg(μA(x))),μB(x)) | min(min(μA(x),sg(νA(x))),νB(x)) | ||
| →101 | max(min(νA(x),sg(μA(x))),min(μB(x),sg(νB(x)))) | min(min(μA(x),sg(νA(x))),min(νB(x),sg(μB(x)))) | ||
| →102 | max(νA(x),min(μB(x),sg(νB(x)))) | min(μA(x),min(νB(x),sg(μB(x)))) | ||
| →103 | max(min(1-μA(x),sg(μA(x))),1-νB(x)) | min(μA(x),sg(1-μA(x)),νB(x)) | ||
| →104 | max(min(1-μA(x),sg(μA(x))),min(1-νB(x),sg(νB(x)))) | min(min(μA(x),sg(1-μA(x))),min(νB(x),sg(1-νB(x)))) | ||
| →105 | max(1-μA(x),min(1-νB(x),sg(νB(x)))) | min(μA(x),min(νB(x),sg(1-νB(x)))) | ||
| →106 | max(min(νA(x),sg(1-νA(x))),μB(x)) | min(min(1-νA(x),sg(νA(x))),1-μB(x)) | ||
| →107 | max(min(νA(x),sg(1-νA(x))),min(μB(x),sg(1-μB(x)))) | min(min(1-νA(x),sg(νA(x))),min(1-μB(x),sg(μB(x)))) | ||
| →108 | max(νA(x),min(μB(x),sg(1-μB(x)))) | min(1-νA(x),min(1-μB(x),sg(μB(x)))) | ||
| →109 | νA(x)+min(sg(1-μA(x)),μB(x)) | μA(x).νA(x)+min(sg(1-μA(x)),νB(x)) | ||
| →110 | max(νA(x),μB(x)) | min(μA(x).νA(x)+sg(1-μA(x)),νB(x)) | ||
| →111 | max(νA(x),μB(x).νB(x)+sg(1-μB(x))) | min(μA(x).νA(x)+sg(1-μA(x)),νB(x).(μB(x).νB(x)+sg(1-μB(x)))+sg(1-νB(x))) | ||
| →112 | νA(x)+μB(x)-νA(x).μB(x) | μA(x).νA(x)+sg(1-μA(x)).νB(x) | ||
| →113 | νA(x)+(μB(x).νB(x)-νA(x).(μB(x).νB(x)+sg(1-μB(x))) | (μA(x).νA(x)+sg(1-μA(x))).(νB(x).(μB(x).νB(x)+sg(1-μB(x)))+sg(1-νB(x))) | ||
| →114 | 1-μA(x)+min(sg(1-μA(x)),1-νB(x)) | μA(x).(1-μA(x))+min(sg(1-μA(x)),νB(x)) | ||
| →115 | 1-min(μA(x),νB(x)) | min(μA(x)(1-μA(x))+sg(1-μA(x)),νB(x)) | ||
| →116 | max(1-μA(x),(1-νB(x)).νB(x)+sg(νB(x))) | min(μA(x).(1-μA(x))+sg(1-μA(x)),νB(x).((1-νB(x)).νB(x)+sg(νB(x)))+sg(1-νB(x))) | ||
| →117 | 1-μA(x)-νB(x)+μA(x).νB(x) | (μA(x).(1-μA(x))+sg(1-μA(x))).νB(x) | ||
| →118 | (1-μA(x)).sg(νB(x))+μA(x).νB(x).(1-νB(x)) | (μA(x)-μA(x)2+sg(1-μA(x))).((1-νB(x)).νB(x)2+sg(1-νB(x)))+sg(1-νB(x))(x)) | ||
| →119 | νA(x)+min(sg(νA(x)),μB(x)) | (1-νA(x)).νA(x)+min(sg(νA(x)),1-μB(x)) | ||
| →120 | max(νA(x),μB(x)) | min((1-νA(x)).νA(x)+sg(νA(x)),1-μB(x)) | ||
| →121 | max(νA(x),μB(x).(1-μB(x))+sg(1-μB(x))) | min((1-νA(x)).νA(x)+sg(νA(x)),(1-μB(x)).(μB(x).(1-μB(x))+sg(1-μB(x)))+sg(μB(x))) | ||
| →122 | νA(x)+μB(x)-νA(x).μB(x) | ((1-νA(x)).νA(x)+sg(νA(x))).(1-μB(x)) | ||
| →123 | νA(x)+μB(x).(1-μB(x)-νA(x).(μB(x).(1-μB(x))+sg(1-μB(x))) | ((1-νA(x)).νA(x)+sg(νA(x))).(((1-μB(x)).(μB(x).(1-μB(x))+sg(1-μB(x))))+sg(μB(x))) | ||
| →124 | μB(x)+min(sg(1-νB(x)),νA(x)) | νB(x).μB(x)+min(sg(1-νB(x)),μA(x)) | ||
| →125 | max(μB(x),νA(x)) | min(νB(x).μB(x)+sg(1-νB(x)),μA(x)) | ||
| →126 | max(μB(x),νA(x).μA(x)+sg(1-νA(x))) | min(νB(x).μB(x)+sg(1-νB(x)),μA(x).(νA(x).μA(x)+sg(1-νA(x)))+sg(1-μA(x))) | ||
| →127 | μB(x)+νA(x)-μB(x).νA(x) | (νB(x).μB(x)+sg(1-νB(x))).μA(x) | ||
| →128 | μB(x)+νA(x).μA(x)-μB(x).(νA(x).μA(x)+sg(1-νA(x))) | (νB(x).μB(x)+sg(1-νB(x))).(μA(x).(νA(x).μA(x)+sg(1-νA(x)))+sg(1-μA(x))) | ||
| →129 | 1-νB(x)+min(sg(1-νB(x)),1-μA(x)) | νB(x).(1-νB(x))+min(sg(1-νB(x)),μA(x)) | ||
| →130 | 1-min(νB(x),μA(x)) | min(νB(x).(1-νB(x))+sg(1-νB(x)),μA(x)) | ||
| →131 | max(1-νB(x),(1-μA(x)).μA(x)+sg(μA(x))) | min(νB(x).(1-νB(x))+sg(1-νB(x)),μA(x).((1-μA(x)).μA(x)+sg(μA(x)))+sg(1-μA(x))) | ||
| →132 | 1-μA(x).νB(x) | (νB(x).(1-νB(x))+sg(1-νB(x))).μA(x) | ||
| →133 | 1-νB(x)+(1-μA(x)).μA(x)-(1-νB(x)).((1-μA(x)).μA(x)+sg(μA(x))) | (νB(x).(1-νB(x))+sg(1-νB(x))).(μA(x).((1-μA(x)).μA(x)+sg(μA(x)))+sg(1-μA(x))) | ||
| →134 | μB(x)+min(sg(μB(x)),νA(x)) | (1-μB(x)).μB(x)+min(sg(μB(x)),1-νA(x)) | ||
| →135 | max(μB(x),νA(x)) | min((1-μB(x)).μB(x)+sg(μB(x)),1-νA(x)) | ||
| →136 | max(μB(x),νA(x).(1-νA(x))+sg(1-νA(x))) | min((1-μB(x)).μB(x)+sg(μB(x)),(1-νA(x)).(νA(x).(1-νA(x))+sg(1-νA(x)))+sg(νA(x))) | ||
| →137 | μB(x)+νA(x)-μB(x).νA(x) | ((1-μB(x)).μB(x)+sg(μB(x))).(1-νA(x)) | ||
| →138 | μB(x)+νA(x).(1-νA(x))-μB(x). |
((1-μB(x)).μB(x)+sg(μB(x))).(1-νA(x).(νA(x).(1-νA(x)) + sg(1-νA(x)) + sg(νA(x)) | ||
| →139 | (νA(x) + μB(x))/2 | (μA(x) + νB(x))/2 | ||
| →140 | (νA(x) + μB(x) + min(νA(x), μB(x)))/3 | (μA(x) + νB(x) + max(μA(x), νB(x)))/3 | ||
| →141 | (νA(x) + μB(x) + max(νA(x), μB(x)))/3 | (μA(x) + νB(x) + min(μA(x), νB(x)))/3 | ||
| →142 | (3 - μA(x) - νB(x) - max(μA(x), νB(x)))/3 | (μA(x) + νB(x) + max(μA(x), νB(x)))/3 | ||
| →143 | (1 - μA(x) + μb(x) + min(1 - μA(x), μB(x)) )/3 | (2 + μA(x) - μB(x) - min(1 - μA(x), μB(x)))/3 | ||
| →144 | (1 + νA(x) - νb(x) + min(νA(x), 1 - νB(x)) )/3 | (2 - νA(x) + νB(x) + min(νA(x), 1 - νB(x)))/3 | ||
| →145 | (νA(x) + μB(x) + min(νA(x), μB(x)))/3 | (3 - νA(x) - μB(x) - min(νA(x), μB(x)))/3 | ||
| →146 | (3 - μA(x) - νB(x) - min(μA(x), νB(x)))/3 | (μA(x) + νB(x) + min(μA(x), νB(x)))/3 | ||
| →147 | (1 - μA(x) + μb(x) + max(1 - μA(x), μB(x)) )/3 | (2 + μA(x) - μb(x) - max(1 - μA(x), μB(x)) )/3
|
References
- On some properties of intuitionistic fuzzy implications, Michał Baczyński, 2003
- Intuitionistic fuzzy implications and Modus Ponens, Krassimir Atanassov, 2005
- A property of intuitionistic fuzzy implications, Yun Shi and Violeta Tasseva, 2005
- On some intuitionistic fuzzy implications, Krassimir Atanassov, 2006
- On a new intuitionistic fuzzy implication of Gaines-Rescher's type, Beloslav Riečan, Krassimir Atanassov, 2007
- A study on some intuitionistic fuzzy implications, Violeta Tasseva, Desislava Peneva, 2007
- On intuitionistic fuzzy subtraction, generated by an implication from Kleene-Dienes type, Lilija Atanassova, 2009
- A new intuitionistic fuzzy implication, Lilija Atanassova, 2009
- Intuitionistic fuzzy implications and axioms for implications, Krassimir Atanassov and Dimitar Dimitrov, 2010
- Four modal forms of intuitionistic fuzzy implication →@ and two related intuitionistic fuzzy negations. Part 1, Lilija Atanassova, 2010
- Some Remarks about L. Atanassova’s Paper “A New Intuitionistic Fuzzy Implication”, Piotr Dworniczak, 2010
- On the basic properties of the negations generated by some parametric intuitionistic fuzzy implications, Piotr Dworniczak, 2011
- On some two-parametric intuitionistic fuzzy implications, Piotr Dworniczak, 2011
- Second Zadeh's intuitionistic fuzzy implication, Krassimir Atanassov, 2011
- "What Links Here" References
- Issue:Four modal forms of intuitionistic fuzzy implication →@ and two related intuitionistic fuzzy negations. Part 1 (← links)
- Issue:On some two-parametric intuitionistic fuzzy implications (← links)
- Issue:Automatic verification of properties of intuitionistic fuzzy connectives via Mathematica (← links)
- Issue:A property of the intuitionistic fuzzy implications (← links)
- Issue:On two modifications of the intuitionistic fuzzy implication →@ (← links)
- Issue:On the intuitionistic fuzzy form of the classical implication (A → B) ∨ (B → A) (← links)
- Issue:Properties of the intuitionistic fuzzy implication →187 (← links)
- Issue:On a special type of intuitionistic fuzzy implications (← links)
- Issue:Properties of the intuitionistic fuzzy implication →189 (← links)
- Issue:Intuitionistic fuzzy implication →190 (← links)
- Issue:Properties of the intuitionistic fuzzy implication →188 (← links)
