# Complement of Set

The complement of set A, denoted by A’ , is the set of all elements in the universal set that are not in A. It is denoted by A’ Some Properties of Complement Sets

1) A ∪ A′ = U
2) A ∩ A′ = Φ
3) Law of double complement : (A′ )′ = A
4) Laws of empty set and universal set Φ′ = U and U′ = Φ.

Examples :

1) If A = { 1, 2, 3, 4} and U = { 1, 2, 3, 4, 5, 6, 7, 8} then find A complement ( A’).

Solution :
A = { 1, 2, 3, 4} and Universal set = U = { 1, 2, 3, 4, 5, 6, 7, 8}

Complement of set A contains the elements present in universal set but not in set A.

Elements are 5, 6, 7, 8.

∴ A complement = A’ = { 5, 6, 7, 8}.

2) If B = { x | x is a book on Algebra in your library} . Find B’.

Solution : B’ = { x | x is a book in your library and x ∉ B }

3) If A = { 1, 2, 3, 4, 5 } and U = N , then find A’.

Solution :
A = { 1, 2, 3, 4, 5 }

U = N

⇒ U = { 1, 2, 3, 4, 5, 6, 7, 8, 9,10,… }

A’ = { 6, 7, 8, 9, 10, … }

4) If A = { x | x is a multiple of 3, x ∉ N }. Find A’.

Solution :
As a convention, x ∉ N in the bracket indicates N is the universal set.

N = U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11, … }

A = { x | x is a multiple of 3, x ∉ N }

A = { 3, 6, 9, 12, 15, … }

So, A’ = { 1, 2, 4, 5, 7, 8, 10,11, … }

Set Theory

Sets
Representation of Set
Cardinal Number
Types of Sets
Pairs of Sets
Subset
Complement of Set
Union of the Sets
Intersection of Sets
Operations on Sets
De Morgan's Law
Venn Diagrams
Venn-diagrams for sets
Venn-diagrams for different situations
Problems on Intersection of Two Sets
Problems on Intersection of Three Sets