(i) Equality of two fuzzy sets
(ii) Containment
(iii) a - cut and a - level set
(iv) union of two fuzzy sets
(v) intersection of two fuzzy sets
(vi) union of two i - v fuzzy sets

Ans.

(i). Equality of two fuzzy sets :

Let A and B be two fuzzy sets of X(¹f) with membership functions mA and mB. We say that A and B are equal written by A = B, if and only if

mA(x) = mB(x) " x Î X

Example :

Suppose X = {1, 2, 3}. Consider the fuzzy sets A, B, C of X given by

A = {1, 2, 3}

2 7 0

B = {2, 1, 3}, and

4 2 0

B = {2, 3, 1},

7 0 2

and A = C, A ¹ B, B ¹ C

(ii). Containment :

Let X be a set (¹f) and A, B are two fuzzy sets of X with membership function mA and mB respectively. We say that the fuzzy set A is contained in the fuzzy set B if and only if

mA(x) = mB(x)" x Î X.

We may say, in another terminology, that a is a fuzzy subset of B, denoted by AÍB.

Example :

If X = {1, 2, 3} and A, B, C are three fuzzy sets given by

A = {.1, .5, .1}, B = {.1, .4, .9}

1 2 3 1 2 3

C = {.1, .6, .1}, then B Í A, C Ë A,

1 2 3

(iii). a - cut and a - level set

The a - cut and a - level set of a fuzzy set a of the set X is the following crisp (i.e. conventional) set given by

Aa {xÎX:mA(x) a}

Example : Suppose X = {1, 2, 3}. Consider the fuzzy set A of given by

A = {.5, .1, .1, .8}

2 1 4 3

clearly, A1 = {1, 2, 3, 4},

A6 = {3, 4}, A0 = {1, 2, 3, 4}

A1 = { 4}, A3 = f

Clearly, if a ³ b, Aa Í Ab write A È B, is defined as

mAÈ B (x)B(x) = mA(x) V mB(x), "x Î X,

where "V" is the 'maximum' operator.

Example:

If x = {1, 2, 3, 4}, A = .2 + .5 + .8 + .1, B = .1 + .8 + .5 + .2 then

1 2 3 4 1 2 3 4

A È B = {(x, mAÈB(x))}, where

mAÈB(x) = [mLAÈB(x), mÈAÈB(x)], xÎX.

where

mLAÈB(x) = max[mLA(x), mLB(x)],

mÈAÈB(x) = max[mÈA(x), mÈB(x)], " x Î X.

This definition may be generalized, in a natural way, to the case of a union of n i-v fuzzy sets.

Cont...