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.
2. Intersection
The Intersection of sets A and B is the set of elements which are common to A and B, that is, those elements which belong to A and which belong to B. We denote the intersection of A and B by:
A∩B
Which is read “A intersection B”.
Example 2.1: In the Venn diagram in fig 2.2, we have shaded A∩B, the area that is common to both A and B
Example 2.2: Let S = {a, b, c, d} and T = {f, b, d, g}. Then S∩T = {b, d}
Example 2.3: Let V = 2, 3, 6, ......} i.e. the multiples of 2; and
Let W = {3, 6, 9,....} i.e. the multiples of 3. Then
V∩W = {6, 12, 18......}
The intersection of A and B may also be defined concisely by
A∩B = {x∈A, x∈B}
Here, the comma has the same meaning as “and”.
Remark 2.3: It follows directly from the definition of the intersection of two
sets that;
A∩B =B∩A
Remark 2.4:
Each of the sets A and B contains A∩B as a subset, i.e.,
(A∩B)⊂A and (A∩B) ⊂B
Remark 2.5: If sets A and B have no elements in common, i.e. if A and B are disjoint, then the intersection of A and B is the null set, i.e. A∩B =∅
In some books, especially on probability, the intersection of A and B is denoted by AB and is called the set-theoretic product of A and B or, simply, A times B.