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.
1. Union
The union of sets A and B is the set of all elements which belong to A or to B or to both. We denote the union of A and B by;
A∪B
Which is usually read “A union B”
Example 1.1: In the Venn diagram in fig 2-1, we have shaded A∪B,
i.e. the area of A and the area of B.
Example 1.2: Let S = {a, b. c. d} and T = {f, b, d, g}.
Then S. T = {a, b, c, d, f, g}.
Example 1.3: Let P be the set of positive real numbers and let Q be the set of negative real numbers. The P∪Q, the union of P and Q, consist of all the real numbers except zero. The union of A and B may also be defined concisely by:
A∪B = {xx∈A or x∈B}
Remark 2.1: It follows directly from the definition of the union of two sets that A∪B and B∪A are the same set, i.e.,
A∪B = B∪A
Remark 2.2: Both A and B are always subsets of A and B that is,
A⊂ (A∪B) and B⊂ (A∪B)
In some books, the union of A and B is denoted by A + B and is called the settheoretic sum of A and B or, simply, A plus B