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Complement of Set

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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

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