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.
4. Complement
The complement of a set A is the set of elements that do not belong to A, that is, the difference of the universal set U and A. We denote the complement of A by A′
Example 4.1: In the Venn diagram in Fig 2.4, we shaded the complement of A, i.e. the area outside A. Here we assume that the universal set U consists of the area in the rectangle.
A’ is shaded
Fig. 2.4
Example 4.2: Let the Universal set U be the English alphabet and let T = {a, b, c}. Then;
T’ = {d, e, f, ....., y, z}
Example 4.3:
Let E = {2, 4, 6, ...}, that is, the even numbers.
Then E′ = {1, 3, 5, ...}, the odd numbers. Here we assume that the universal set is the natural numbers, 1, 2, 3,.....
The complement of A may also be defined concisely by;
A′= {x|x∈U, x∉A} or, simply,
A′= {x|x∉A}
We state some facts about sets, which follow directly from the definition of the complement of a set.
Remark 2.8: The union of any set A and its complement A′ is the universal set, i.e.,
A∪A’ = U
Furthermore, set A and its complement A′ are disjoint, i.e.,
A∩A’ = ∅
Remark 2.9: The complement of the universal set U is the null set ∅, and vice versa, that is,
U’ = ∅ and ∅’ = U
Remark 2.10: The complement of the complement of set A is the set A itself. More briefly,
(A′)′ = A
Our next remark shows how the difference of two sets can be defined in terms of the complement of a set and the intersection of two sets. More specifically,
we have the following basic relationship:
Remark 2.11: The difference of A and B is equal to the intersection of A and the complement of B, that is,
A – B = A∩B′
The proof of Remark 2.11 follows directly from definitions:
A – B = {x|x∈A, x∉B} = {x|x∈A, x∉B’} = A∩B’