8.6.2. Logical Operations of Ordinary Sets

 

Definition 1. Given A, B Π2W ,

A » B = {w‘w Œ A or w Œ B},

A « B = {w‘w Œ A and w Œ B},

 = {w‘w œ A},

are called the union of set A and B, the intersection of A and B and the complement of A, respectively. When "»", "«", and "-" operators are used in combination, "-" has higher priority than "»" and "«".

It can be proven that for any W and A, B Π2W , the following relationships hold:

( ) =  «  ,

( ) =  »  .

These are called De-Morgan's law.

The following are some properties for set arithmetics.

A » A = A , A « A = A

A » B = B » A , A « B = B « A

(A » B)»C = A»(B » C )

(A « B)«C = A«(B « C )

(A « B)»B = B,

(A » B)«B = B

A«(B » C) = (A « B)»(A « C),

A»(B « C) = (A » B)«(A » C)

A » W = W, A « W = A

A » f = A, A « f = f

( ) = A

A »  = W

A «  = f .

Definition 2. The two denotations

 Ai = {w‘wŒW, $ iŒI such that wŒAi }

 Ai = {w‘wŒW, $ iŒI such that wŒAi }

are called the union and intersection of set class { Ai‘iŒI } .

I = { 1, 2, ... , n, ... } is called index set.

When I = { 1, 2 }, definition 2 is equivalent to definition 1.

Definition 3.

A - B = {w‘wŒA and wœB } is called the difference set of B for A.

A - B = A «  

 = » - A

A projection from W to F is defined by:

f : W Æ F .

Projection is the extension to the concept of a function. For any w ΠW, there exists an element j = f(w). w is the original image and f(w) is called the image of w.

W is the definition range for f, and

f(W) = { j‘$ wŒW such that j = f(w) } .

f(W) is called the value range.

If f(W) = F , then f is full projection from W to F.

If for any given w1, w2 ΠW and w1 _ w2, we have

f(w1) _ f(w2) ,

then f is a one to one projection.

Definition 4. Given A Π2W , determine a projection from W to { 0, 1 },

XA : W Æ { 0, 1 } such that

XA(w) =  

XA is the characteristic function of set A.

The value of the characteristic function of A at w is XA (w). X(w) is called the degree of membership for w in A.

Obviously, when w Œ A, the degree of membership for w belonging to A is 1 indicating that w is absolutely an element of A. When w œ A, the degree of membership becomes 0, indicating that w does not belong to A at all.