FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 2.3- Sub-set :
A set N is a subset of a set X, if all the elements of N are contained in/members of the larger set X.
Example
If, X = {3, 5, 6, 8, 9, 10, 11, 13}
And, N = {5, 11, 13}
Then, N is a subset of X.
That is, N ⊆ X (where ⊆ means ‘is a subset of’).
Number of Subsets
If, M = {a, b, c}
Then, the subsets of M are:
{a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}, {}
Therefore, the number of subsets, S = 8
And the formula, S = 2n
Where,
S is the number of sets
And, n is the number of elements of the set
in the formula used to calculate the number of subsets of a given set.
So from above, M = {a, b, c}
S = 2n
= 2^3
= 2 x 2 x 2
= 8
Note: Every set is a subset of itself, and the empty set is a subset of all sets. - Proper Sub-set :
If each and every element of a set A are the elements of B and there exists at least one element of B
that does not belongs to A, then the set A is said to be a proper sub-set of B (or B is called super-set of
A). Symbolically, we may write,
A ⊂ B (read as A is proper sub-set of B)
And B ⊂ A means A is a super-set of B.
If B = {a, b, c}, then proper sub-sets are {a}, {b}, {c}, {a, b}, {b,c}, {a, c}, φ
[Note : (i) A set is not proper sub-set of itself.
(ii) Number of proper sub-sets of a set A containing n elements is 2n –1
(iii) φ is not proper sub-set of itself].