Page 77 - ICSE Math 8

Proper subset
A set A is said to be a proper subset of B, if:
(a) all the elements of set A are in set B.
(b) there is at least one element in set B which is not in set A.
If A is a proper subset of B then, it is denoted by A ⊂ B.
For example, let A = {k, l, m, n} and B = {k, l, m, n, o}. Here all the elements k, l, m, n of set A are in set B
and there is one element ‘o’ in B which is not in set A. So, A is proper subset of B or A ⊂ B.

Properties of a subset
(a) Every set is a subset of itself, i.e., for any set A, A ⊂ A.
(b) Null set is a subset of every set, i.e., for any set A, φ ⊂ A.
(c) A ⊂ B ⇒ either A ⊂ B or A = B
(d) If A ⊂ B and B ⊂ C ⇒ A ⊂ C
(e) If A ⊂ B and B ⊂ A ⇒ A = B
Example 6: If A = {p, q, r, s, t} and C = {p, q, m}. Can we say that C is a subset of A?

Solution: Though p and q belong to set A, m ∉ A. So, all the elements of set C are not in set A.
Hence C is not a subset of A.
Number of subsets of a set
Consider a set A = {a, b}.
Subsets of A are φ, {a}, {b}, {a, b}. So, there are 4 subsets of set A. Now observe the following table.

Number of
Set Subsets Number of subsets
elements in set
A = {a} 1 φ, {a} 2 = 2 1
B = {a, b} 2 φ, {a}, {b}, {a, b} 4 = 2 2
φ, {a}, {b}, {c}, 3
C = {a, b, c} 3 8 = 2
{a, b}, {a, c}, {b, c}, {a, b, c}

n
Thus, if there are n elements in a set, then there are 2 number of subsets of that set.
Point to remember
n
Number of proper subsets of a set having n elements is 2 – 1.
Power set
A set of subsets of a given set is called the power set of the given set. For a set X, its power set is denoted by
2
P(X). For example, in the above table, n[P(A)] = 2, n[P(B)] = 4 = 2 , n[P(C)] = 8 = 2 3
n
So, number of elements in the power set of a set having n elements is 2 .
Example 7: Write down all the subsets of the set P = {–1, 0, 1}.
Solution: Subsets of P are φ, {–1}, {0}, {1}, {–1, 0}, {–1, 1}, {0, 1}, {–1, 0, 1}.

Example 8: If a set has 64 subsets, how many elements are there in the set?
Solution: Let the number of elements in the set be n.
Number of subsets = 2 n
n
6
2 = 64 = 2
⇒ n = 6
Hence, the number of elements in the set is 6.

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