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 = 2
n

So , number of sub-sets of set A = 2
4

= 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 2
n - 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 2
n .

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 2
2 = 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 2
3 = 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 ξ or 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.

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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