SET OPERATIONS
In arithmetic, we learn to add, subtract and multiply, that is, we assign to each pair of numbers x and y a number x + y called the sum of x and y, a number x – y called the difference of x and y, and a number xy called the product of x and y.
These assignments are called the operations of addition, subtraction and multiplication of numbers. In this unit, we define the operation Union, Intersection and difference of sets, that is, we will assign new pairs of sets A and B. In a later unit, we will see that these set operations behave in a manner some what similar to the above operations on numbers.
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