Page 78 - ICSE Math 8

Universal set
The largest possible set which contains all the given sets as its subsets is called a universal set. It is denoted
by U or ξ. Besides the elements of the given sets, a universal set may also have extra elements.
For example, if A = {2, 3, 5, 8, 13}, B = {1, 5, 7, 9, 12}, C = {2, 4, 6, 8, 10, 11, 14} and
U = {1, 2, 3, 4,.........15}, then A ⊆ U, B ⊆ U and C ⊆ U.
Points to remember

• Universal set is the largest superset.
• Universal set is not unique.

EXERCISE 6.2

1. Which of the following are examples of null set?
(a) Set of even prime numbers (b) {x : x is a natural number < 1}
(c) {x : x is the 30th day in the month of February}
2. Write whether the given sets are finite or infinite.
(a) Set of stars in the sky (b) Set of integers
(c) {x : x is a consonant of English alphabet} (d) Set of students in your school
3. If A = {1, 2, 3, 4, 5, 6, 7}, then which of the following are true?

(a) 3 ∈ A (b) 2 ⊆ A (c) {5, 7} ⊆ A
(d) 6 ∈ A (e) 7 ⊆ A (f) φ ∈ A
4. Write all the subsets of the following sets.
(a) A = {p, q} (b) B = {–3, 2, 4} (c) C = {–8} (d) D = {–1, 0, 2, 5}
5. If U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then which of the following sets are subsets of U?
(a) A = {1, 2, 5, 9} (b) B = {8, 0, 6, 7, 10} (c) C = {2, 3, 4, 8, 11}
(d) D = {1, 7, 9, 10, 8, 12} (e) E = {3, 5, 9, 7}
6. Write the cardinal number of each of the following sets.
(a) A = {p, q, r, s, t} (b) P = {18, 20, 22, 24, 26, 28, 30}

(c) B = {x : x = 2y – 3, x, y ∈ Z, 1 ≤ y < 5}
7. Consider the sets L = {a, c, d, e}, M = {b, d, e, f, g} and N = {a, g, h}. Which of the following may be
considered as universal set(s) for all the three sets L, M and N?

(a) {a, b, c, d, e, f, g, h} (b) {b, c, d, e, f, g, h, i}
(c) {a, b, c, d, e, f, g, h, i} (d) φ

Operations on Sets
Union of sets
Let A and B be any two sets. The union of sets A and B is the set of all the elements which belong to either A
or B or both. Elements common to both the sets are taken only once in the union set. It is denoted by A ∪ B
(read as A union B). For example, if A = {2, 3, 4, 5} and B = {3, 5, 7, 9, 11}, then A ∪ B = {2, 3, 4, 5, 7, 9, 11}.

Elements 3 and 5 are in both A and B, but we write them only once as elements in a set cannot be repeated.

Properties of union of sets

(a) Union of sets is commutative, i.e., for any two sets A and B, A ∪ B = B ∪ A.
(b) Union of sets is associative, i.e., for any three sets A, B and C, A ∪ (B ∪ C) = (A ∪ B) ∪ C.

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