Talk:Fuzzy set operations

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): $\min \left(\mu _{1}\left(x\right),\mu _{2}\left(x\right)\right)$
• Union (OR) 1 (Standard union): $\max \left(\mu _{1}\left(x\right),\mu _{2}\left(x\right)\right)$
• Intersection (AND) 2 (Bounded difference): $\max \left(0,\mu _{1}\left(x\right)+\mu _{2}\left(x\right)-1\right)$
• Union (OR) 2 (Bounded sum): $\min \left(1,\mu _{1}\left(x\right)+\mu _{2}\left(x\right)\right)$
• 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: $\max \left(0,\mu _{1}\left(x\right)-\mu _{2}\left(x\right)\right)$
• 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: ${\begin{cases}\mu _{1},&\mu _{2}=1\\\mu _{2},&\mu _{1}=1\\0,otherwise\end{cases}}$
• Drastic union: ${\begin{cases}\mu _{1},&\mu _{2}=0\\\mu _{2},&\mu _{1}=0\\1,otherwise\end{cases}}$

Solarapex 15:25, 30 July 2007 (UTC)

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