Properties of Complement of a Set in Set Theory

In Maths, sets can be defined as a collection of well-defined objects or elements. A set can be represented by a capital letter symbol, and the number of elements in the finite set can be represented as the cardinal number of a set in a curly bracket {…}.

For example, set A is a collection of all the natural numbers, such as A equals {1, 2, 3, 4, 5, 6, 7, 8, ….. ∞}.

Sets can be represented in three forms:

1. Roster Form: Example - Set of even numbers less than 8 = {2, 4, 6}.

2. Statement Form: Example: A = {Set of Odd numbers less than 9}.

3. Set Builder Form: Example: A = {x: x=2n, n ∈ N and 1 ≤ n ≤ 4}.

In this article, we are going to discuss the properties of the complement of a set, we are going to go through the properties of the complement of a set in brief.

What are the Types of Sets?

A set has many types, such as;

1. Empty Set or Null Set: It has no element present in it. Example: A = {} is a null set.

2. Finite Set: It has a limited number of elements. Example: A = {1, 2, 3, 4}.

3. Infinite Set: It has an infinite number of elements. Example: A = {x: x is the set of all whole numbers}.

4. Equal Set: Two sets which have the same members. Example: A = {1, 2, 5} and B = {2 , 5, 1}: Set A = Set B.

5. Subsets: A set ‘A’ is said to be a subset of B if each element of A is also an element of B. Example: A = {1, 2}, B = {1, 2, 3, 4}, then A ⊆ B.

6. Universal Set: A set that consists of all elements of other sets present in a Venn diagram. Example: A = {1, 2}, B = {2, 3}, The universal set here will be, U = {1, 2, 3}.

Properties of Complement of Set

There are three properties of the complement of a set. Let’s go through these three properties of the complement of a set:

1. Complement Laws: This is the first of the three properties of the complement of a set. The union of a set A and its complement denoted by A’ gives the universal set U of which A and A’ are a subset.

A ∪ A’ equals U

Also, the intersection of a set A and its complement A’ gives the empty set denoted by ∅.

A ∩ A’ = ∅

For Example: If  U = {1 , 2 , 3 , 4 , 5 }  and  A = {1 , 2 , 3} then  A’ = {4 , 5}. From this, it can be seen that A ∪ A’ = U = { 1 , 2 , 3 , 4 , 5}

Also, A ∩ A’ = ∅

2. Law of Double Complementation:

This is the second of the three properties of the complement of a set. According to this law of Double Complementation, if we take the complement of the complemented set named A’ then, we get the set A itself.

(A’ )’ equals A

In the previous example we can see that, if U = {1, 2, 3, 4, 5} and A = {1, 2, 3}  then A’ = {4 , 5} . Now if take the complement of set A’ we get the following,

(A’ )’ = {1, 2, 3} = A , This gives us the set A itself.

3. Law of Empty Set and Universal Set:

According to this law, the complement of the universal set gives us the empty set and vice-versa that is,

∅’ equals U And U equals ∅’

These three are the properties of the complement of a set. These properties of the complement of a set are useful in Mathematics.

Solved Examples

Question 1) A universal set named U which consists of all the natural numbers which are multiples of the number 3, less than or equal to the number 20. Let set A be a subset of U which consists of all the even numbers and set B is also a subset of U consisting of all the prime numbers. Verify De Morgan Law.

Solution) We have to verify (A ∪ B)’ equals A’ ∩ B’ and (A ∩ B)’ equals A’∪B’. Given that, Using the properties of the complement of a set, let’s solve.

U equals {3, 6, 9, 12, 15, 18}

A equals {6, 12, 18}

B equals {3}

The union of both A and B can be given as,

A ∪ B equals {3, 6, 12, 18}

The complement of this union is given by,

(A ∪ B)’ equals {9, 15}

Also, the intersection and its complement are given by:

A ∩ B = ∅

(A ∩ B)’ equals {3, 6, 9, 12, 15,18}

Now, the complement of the set A and set B can be given as:

A’ = {3, 9, 15}

B’ = {6, 9, 12, 15, 18}

Taking the union of both these sets, we get,

A’∪B’ = {3, 6, 9, 12, 15, 18}

And the intersection of the complemented sets can be given as,

A’ ∩ B’ = {9, 15}

We can see that:

(A ∪ B)’ = A’ ∩ B’ = {9, 15}

And also,

(A ∩ B)’ = A’ ∪ B’ = {3, 6, 9, 12, 15,18}

Hence, the above-given result is true in general and is known as the De Morgan Law.