Talk:Fuzzy set operations

Template:WikiProject banner shell should the subidempotency axiom for the intersection be "i(a,a)<= a" rather than "i(a,a) < a" ?

Although in mathematics the prefix 'sub-' is usually understood in a non-strict way (a set is a subset of itself, a group is a subgroup of itself, and so on) in this case the property "i(a,a)<=a" already follows from the other axioms, since "i(a,a)<=i(a,1)=a", so only strict subidempotency is a non-trivial property. It is standard to call the property "i(a,a)<a" either subidempotency (as done in the Klir-Yuan textbook) or Archimedeanity, however it must be noted that it is not equivalent to the ordinary meaning of the Archimedean property (i.e. the iteration of any non-trivial "a" by the operation eventually yields a number as close to 0 as wished) unless further assumptions on "i" are imposed (e.g. continuity). Of course, for "u" the above should read `as close to 1 as wished'.

I think it would be important to note which of the axioms are `true' axioms (those defining a triangular norm and a triangular conorm) and which are in fact additional requierements which may or may not be imposed. Otherwise this mixed usage of the word `axiom' will be confusing to any unaware reader. For instance, the minimum is called the standard intersection but it fails the subidempotency `axiom'.--155.210.235.125 17:50, 5 July 2006 (UTC)

Other fuzzy set operations

Here's what I found in the book I have (Pospelov, 1986)

• Absolute complement (NOT): $1-{\mu }_1\left(x\right)$
• Intersection (AND) 1 (Standard intersection): $\min ({\mu }_1\left(x\right),{\mu }_2\left(x\right))$
• Union (OR) 1 (Standard union): $\max ({\mu }_1\left(x\right),{\mu }_2\left(x\right))$
• Intersection (AND) 2 (Bounded difference): $\max (0,{\mu }_1\left(x\right)+{\mu }_2\left(x\right)-1)$
• Union (OR) 2 (Bounded sum): $\min (1,{\mu }_1\left(x\right)+{\mu }_2\left(x\right))$
• Intersection (AND) 3 (Algebraic product): ${\mu }_1\left(x\right)\times {\mu }_2\left(x\right)$
• Union (OR) 3 (Algebraic sum): ${\mu }_1\left(x\right)+{\mu }_2\left(x\right)-{\mu }_1\left(x\right)\times {\mu }_2\left(x\right)$
• Relative Complement: $\max (0,{\mu }_1\left(x\right)-{\mu }_2\left(x\right))$
• Concenration (VERY): ${\mu }_1{\left(x\right)}^2$

In other references I saw one more:

• Deconcenration (MORE OR LESS): $\sqrt{{\mu }_1\left(x\right)}$

I'm not sure if this is accurate as it seems English and Russian sources differ on this topic. Solarapex 11:35, 30 July 2007 (UTC)

Added corresponding terms I found in English references.

A couple more:

• Drastic intersection: $\{\begin{array}{ll}{\mu }_1,& {\mu }_2=1\\ {\mu }_2,& {\mu }_1=1\\ 0,otherwise\end{array}$
• Drastic union: $\{\begin{array}{ll}{\mu }_1,& {\mu }_2=0\\ {\mu }_2,& {\mu }_1=0\\ 1,otherwise\end{array}$

Solarapex 15:25, 30 July 2007 (UTC)

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