Similarity in fuzzy systems

Journal of Uncertainty Analysis and Applications

Table 3 For translation of relations

R	Conditional relations	Translation (μ_R(u, v))
R ₁	Reichenbach s-implication	1 - μ_A(u)+μ_A(u)*μ_B(v)
R ₂	Kleen-Dienes s-implication	max(1 - μ_A(u), μ_B(v))
R ₃	Lukasiewicz r- and s-implications	min(1 - μ_A(u)+μ_B(v),1)
R ₄	Rescher-Gaines r-implication	$\{\begin{array}{l} 1, & if μ_{A} (u) \leq μ_{B} (v)) \\ 0, & otherwise \end{array}$
R ₅	Godel r-implication	$\{\begin{array}{l} 1, & if μ_{A} (u) \leq μ_{B} (v)) \\ μ_{B} (v), & otherwise \end{array}$
R ₆	Goguen r-implication	$\{\begin{array}{l} min (μ_{B} (v) / μ_{A} (u), 1), & if μ_{A} (u) > 0 \\ 1, & otherwise \end{array}$
R ₇	Standard t-norm	min(μ_A(u), μ_B(v))
R ₈	Bounded sum t-norm	max(μ_A(u)+μ_B(v) - 1,0)
R ₉	Product t-norm	μ_A(u)*μ_B(v)