Implication in Fuzzy Logic

If P, Q, R are Fuzzy propositions defined on sets A, B, C of their respective universe of discourses, then the fuzzy propositions P, Q, R are given by
P : x is A, A ⊂ X
Q : y is B, B ⊂ Y
R : y is C, C ⊂ Y

P → Q : Max [ 1 – T(P), T(Q) ]
If A, then B (in linguistic rule form)

In set theoretic form, it is given by the relation
R = (A X B) V (Ac X Y)

It membership function is given by


for the linguistic “If A then B else C”, the set theoretic form is given by
R = (A X B) V (Ac X C)

Its membership function is given by –


Other forms of Fuzzy Implication:-

μR(x,y) implication

μR(x,y) when μB(y) less than or equal to μA(x)

Correlation minimum or Mamdani’s Implication:-

Correlation minimum or Mamdani's Implication

Łukasiewicz Implication:-

Lukasiewicz Implication

Bounded-Sum Implication:-

Bounded-Sum Implication

Correlation Product Implication:-


Browerian Implication:-


R-SEQ Implication (Standard Sequence Logic):-

R-SEQ Implication or Standard Sequence Logic