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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