SET OPERATIONS

5. OPERATIONS ON COMPARABLE SETS

The operations of union, intersection, difference and complement have simple properties when the sets under investigation are comparable. The following theorems can be proved.

Theorem 2.1: Let A be a subset of B. Then the union intersection of A and B is precisely A, that is,

                       A⊂B implies A∩B = A

Theorem 2.2: Let A be a subset of B. Then the of A and B is precisely B, that is,

                       A⊂B implies A∪B = B

Theorem 2.3: Let A be a subset of B. Then B’ is a subset of A’, that is,

                         A⊂B implies B’⊂A’

We illustrate Theorem 2.3 by the Venn diagrams in Fig 2-5 and 2-6. Notice

how the area of B’ is included in the area of A’.

Theorem 2.4: Let A be a subset of B. Then the Union of A and (B – A) is precisely B, that is,

A⊂B implies A∪(B – A) = B