SET OPERATIONS

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’