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
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.
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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