Subset and its types

Set A is a subset of B, if every element of set A is also an element of set B.

It is denoted by A ⊆ B

It is read as A is a sub-set of B or A is contained in B.

• For A ⊆ B , every element of A must be in B.

Properties of Sub-set

• For A ⊄ B there must be at least one element in A which is not in B.

• Every set is a sub-set of itself. A ⊆ A

• Φ is a sub-set of every set. Φ ⊆ A

• Number of sub-sets of a set A, containing ‘n’ elements is given by 2ⁿ

• If A⊂B and B ⊂C then A⊂C (Transitivity)

• If A ⊂B and B ⊂A then A = B

For example :

1) A = { 2,4,6} and B = { x | x is even natural number }

Solution :
A = { 2, 4, 6}

B = { x | x is even natural number }

B= { 2,4,6,8,10,…}

As every element of set A are in set B

So, A ⊆ B ( A is a sub-set of B)

But B is not a sub-set of A.

2) A = { x| x is a vowel in the word CONFIGURATION } and B = { a,e,i, o,u}

Solution :
A = { x| x is a vowel in the word CONFIGURATION }

A = { o, i, a, u}

B = { a, e, i , o, u}

As all elements of set A are in set B.

∴ A ⊆ B.

3) Write the number of sub-sets of set A = { 4,5,6,7}

Solution :
A = { 4,5,6,7}

Number of elements of A = n = 4

Number of sub-sets is given by the formula = 2ⁿ

So , number of sub-sets of set A = 2⁴

= 16.

Super set

If set A is a sub-set of B, then B is called superset of A. It is denoted by ⊃ .

B ⊃ A and read as B is a super set of A .

For example :

1) A = { 1, 2} , B = { 1, 2, 3 } then A ⊆ B or B ⊃ A [ B contains A ]

So, B is called the superset of A.

2) B = { squares } and D = { Rectangles }

So , B ⊂ D or D ⊃ B.

As A square is always a rectangle.

Proper sub-set

Set A is a proper subset of set B, if every element of set A is an element of set B, and at least one element of B is not an element of A. It is denoted by ⊂

B ⊃ C means : B is a super set of C or B contains C.

• A is not a proper sub-set of itself.

• Number of proper sub-sets of set A, containing ‘n’ elements is 2ⁿ - 1 .

• Φ is not a proper sub-set of itself.

For example :

If N = { set of natural numbers}

W = { set of whole numbers}

Z = { set of integers }

Q = { set of rational numbers }

R = { set of real numbers }

then N ⊆ W ⊆ Z ⊆ Q ⊆ R .

Power Set

The set of all subsets of a set A is the power set of the set A.

It is denoted by P(A).

As it is set of all subsets so it is given by the formula 2ⁿ.

Examples :

1) If A = { 1, 2} then write the power set of A.

Solution :
A = {1, 2}

Number of elements = 2

So, number of power sets will be 2² = 4.

A = P (A) = { {1}, {2}, {1,2}, { } }

2) If B = { 4,8,12} then write the power set of B.

Solution :
B= {4,8, 12}

Number of elements = 3

So, number of power sets will be 2³ = 8.

A = P (A) = { {4}, {8}, {12},{4,8},{8,12}, {4,12},{4,8,12}, { } }

Universal Set

A set which contains all sets under consideration as subset is called an universal set.It is denoted by ξor U

For example :

Let A = { 1,2,3} and B = { 2,3,4}, then the universal set ξ might be { 1,2,3,4,5,6} or {x | x ∈ N } or { x| x ∈ W } or {x | x ∈ N, x ≤ 5 }

Note : The choice of a universal set is not unique.

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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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