8.6.2. Logical Operations of Ordinary Sets
Definition 1. Given A, B 2W ,
A » B = {ww A or w B},
A « B = {ww A and w B},
= {ww 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 2W , 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 = {wwW, $ iI such that wAi }
Ai = {wwW, $ iI such that wAi }
are called the union and intersection of set class { AiiI } .
I = { 1, 2, ... , n, ... } is called index set.
When I = { 1, 2 }, definition 2 is equivalent to definition 1.
Definition 3.
A - B = {wwA and wB } 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 W, 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$ wW 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 W and w1 _ w2, we have
f(w1) _ f(w2) ,
then f is a one to one projection.
Definition 4. Given A 2W , 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 A, the degree of membership for w belonging to A is 1 indicating that w is absolutely an element of A. When w A, the degree of membership becomes 0, indicating that w does not belong to A at all.