In complement of a set if ξ be the universal set and A a subset of ξ,
then the complement of A is the set of all elements of ξ which are not
the elements of A.

Symbolically, we denote the complement of A with respect to ξ as A’.

**For Example;** If ξ = {1, 2, 3, 4, 5, 6, 7}

A = {1, 3, 7} find A'.**Solution:**

We observe that 2, 4, 5, 6 are the only elements of ξ which do not belong to A.

Therefore, A' = {2, 4, 5, 6}**Note:**

The complement of a universal set is an empty set.

The complement of an empty set is a universal set.

The set and its complement are disjoint sets.

**For Example;**

**1.** Let the set of natural numbers be the universal set and A is a set of even natural numbers,

then A' {x: x is a set of odd natural numbers}

**2.** Let ξ = The set of letters in the English alphabet.

A = The set of consonants in the English alphabet

then A' = The set of vowels in the English alphabet.

**3.** Show that;

**(a) The complement of a universal set is an empty set.**

Let ξ denote the universal set, then

ξ' = The set of those elements which are not in ξ.

= empty set = ϕ

Therefore, ξ = ϕ so the complement of a universal set is an empty set.

**(b) A set and its complement are disjoint sets.**

Let A be any set then A' = set of those elements of ξ which are not in A'.

Let x ∉ A, then x is an element of ξ not contained in A'

So x ∉ A'

Therefore, A and A' are disjoint sets.

Therefore, Set and its complement are disjoint sets

Similarly, in complement of a set when U be the universal set and A is a
subset of U. Then the complement of A is the set all elements of U
which are not the elements of A.

Symbolically, we write A' to denote the complement of A with respect to U.

Thus, A' = {x : x ∈ U and x ∉ A}

Obviously A' = {U - A}

**For Example;** Let U = {2, 4, 6, 8, 10, 12, 14, 16}

A = {6, 10, 4, 16}

A' = {2, 8, 12, 14}

We observe that 2, 8, 12, 14 are the only elements of U which do not belong to A.

Some properties of complement sets

(i) A ∪ A' = A' ∪ A = ∪ (Complement law)

(ii) (A ∩ B') = ϕ (Complement law)

(iii) (A ∪ B) = A' ∩ B' (De Morgan’s law)

(iv) (A ∩ B)' = A' ∪ B' (De Morgan’s law)

(v) (A')' = A (Law of complementation)

(vi) ϕ' = ∪ (Law of empty set

(vii) ∪' = ϕ and universal set)

● **Set Theory**

●** Sets**

**● **Subset

**8th Grade Math Practice**

**From Complement of a Set to HOME PAGE**

**Didn't find what you were looking for? Or want to know more information
about Math Only Math.
Use this Google Search to find what you need.**

## New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.