Page 85 - ICSE Math 7

Example 5: Identify the empty set, singleton set and pair set.
(a) X = {x | x + 5 = 4, x ∈ N} (b) Y = {y | 2y + 5 = 11, y ∈ N}
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(c) A = {x | x = 9, x ∈ Z}

Solution: (a) x + 5 = 4 ⇒ x = 4 – 5 = –1
\ There is no natural number which satisfies x + 5 = 4. So, X = f ⇒ n(X) = 0.
Thus, X is an empty set.
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(b) 2y + 5 = 11 ⇒ 2y = 11 – 5 ⇒ y = = 3 Try This
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\ Y = {3} ⇒ n(Y) = 1
Thus, Y is a singleton set. Identify the disjoint and overlapping
pairs of sets.
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(c) x = 9, x ∈ Z (a) A = {4, 8, 12, 16, 20} and

⇒ x = 3, –3 B = {1, 2, 4, 5, 10, 20}
(b) X = {a | a is a prime number} and
\ A = {3, –3} ⇒ n(A) = 2 Y = {b | b is an even number}
Thus, A is a pair set. (c) P = {1, 3, 5, 7, 11, 15} and
Q = {0, 2, 4, 6, 8, 10}
Subsets
If two sets, X and Y, are such that every member of X is also a member of Y, then X is a subset of Y
and it is denoted by X ⊆ Y (read as X is a subset of Y). It can also be said that Y is a superset of X
and is denoted by Y ⊇ X. If all the elements of X are in set Y and there exists at least one member of
Y which is not in X, then X is known as a proper subset of Y and is denoted by X ⊂ Y.

Points to remember

• Every set is a subset of itself, i.e., A ⊆ A. Maths Info
• Every proper subset is a subset but converse is not always true. The empty set f is a subset of
• X ⊆ Y ⇒ either X ⊂ Y or X = Y every set.
• If X ⊆ Y and Y ⊆ X, then X = Y
• If X ⊆ Y and Y ⊆ Z, then X ⊆ Z
• If X ⊆ f, then X = f
• All the subsets except the set itself are proper subsets of a set.

Number of subsets
n
n
If a set has n elements, then it has 2 subsets and 2 – 1 proper subsets.

For example, if X = {a, b}, then all possible subsets of X are f, {a}, {b} and {a, b}. So, there are
2
2
2 = 4 possible subsets. Also, proper subsets of X are f, {a} and {b} which are 3 (2 – 1) in number.
Universal Set

Universal set is a set which contains all the given sets as its subsets
or (under consideration). It is denoted by U or ξ (read pxi). Except
the elements of the given sets, universal set may have extra elements Maths Info
also.
For example, if A = {2, 4, 6, 8, 10} and B = {1, 3, 5, 6}, then the Universal set for the sets under
set {1, 2, 3, 4, 5, 6, 8, 10} is the universal set for sets A and B. consideration is not unique.
In this case, we can also take N, the set of natural numbers as the
universal set.

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