Fuzzy mathematics

Fuzzy mathematics forms a branch of mathematics including fuzzy set theory and fuzzy logic. It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work "Fuzzy sets".
Definition.
A "fuzzy subset" "A" of a set "X" is a function "A": "X" → "L", where "L" is the interval [0, 1]. This function is also called a membership function. A membership function is a generalization of an indicator function (also called a characteristic function) of a subset defined for "L" = {0, 1}. More generally, one can use any complete lattice "L" in a definition of a fuzzy subset "A".
Fuzzification.
The evolution of the fuzzification of mathematical concepts can be broken down into three stages: 
Usually, a fuzzification of mathematical concepts is based on a generalization of these concepts from characteristic functions to membership functions. Let "A" and "B" be two fuzzy subsets of "X". 
The "intersection" "A" ∩ "B" and "union" "A" ∪ "B" are defined as follows: ("A" ∩ "B")("x") = min("A"("x"), "B"("x")), ("A" ∪ "B")("x") = max("A"("x"), "B"("x")) for all "x" in "X". Instead of and one can use t-norm and t-conorm, respectively 
, for example, min("a", "b") can be replaced by multiplication "ab". A straightforward fuzzification is usually based on and operations because in this case more properties of traditional mathematics can be extended to the fuzzy case. 
An important generalization principle used in fuzzification of algebraic operations is a closure property. Let * be a binary operation on "X". The closure property for a fuzzy subset "A" of "X" is that for all "x", "y" in "X", "A"("x"*"y") ≥ min("A"("x"), "A"("y")). Let ("G", *) be a group and "A" a fuzzy subset of "G". Then "A" is a "fuzzy subgroup" of "G" if for all "x", "y" in "G", "A"("x"*"y"−1) ≥ min("A"("x"), "A"("y"−1)). 
A similar generalization principle is used, for example, for fuzzification of the transitivity property. Let "R" be a fuzzy relation on "X", i.e. "R" is a fuzzy subset of "X" × "X". Then "R" is (fuzzy-)transitive if for all "x", "y", "z" in "X", "R"("x", "z") ≥ min("R"("x", "y"), "R"("y", "z")).
Fuzzy analogues.
Fuzzy subgroupoids and fuzzy subgroups were introduced in 1971 by A. Rosenfeld.
Analogues of other mathematical subjects have been translated to fuzzy mathematics, such as fuzzy field theory and fuzzy Galois theory, fuzzy topology, fuzzy geometry, fuzzy orderings, and fuzzy graphs.