Complement of a Set

 

Welcome to class! 

In today’s class, we will be talking about the complement of a set. Enjoy the class!

Complement of Set

Complement of Set classnotes.ng

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}

  1. 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.

  1. 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 the complement of a set when U be the universal set and A is a subset of U. Then the

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.

 

In our next class, we will be talking about Circle and its Properties.  We hope you enjoyed the class.

Should you have any further question, feel free to ask in the comment section below and trust us to respond as soon as possible.

Get more class notes, videos, homework help, exam practice etc on our app [CLICK HERE]

Upgrade your teaching with ready-made & downloadable class notes on our app [CLICK HERE]

Leave a Reply

Your email address will not be published. Required fields are marked *

Don`t copy text!