Page 89 - ICSE Math 7

Example 15: In a school, there are 30 teachers who teach Science or Mathematics. Out of these
15 teachers teach Mathematics and 8 teach both the subjects. How many teachers teach
Science? How many teach only Science?
Solution: Let U = {Set of 30 teachers}, S = {Teachers who teach Science},
M = {Teachers who teach Mathematics}

n(S ∪ M) = 30, n(M) = 15, n(S ∩ M) = 8
n(S ∪ M) = n(S) + n(M) – n(S ∩ M)
⇒ 30 = n(S) + 15 – 8

⇒ n(S) = 30 – 7 = 23
n(only S) = n(S) – n(S ∩ M) = 23 – 8 = 15

Exercise 6.3

1. Find the union and intersection of each of the following pairs of sets.
(a) A = {3, 6, 9}, B = {2, 3, 5, 7}
(b) C = {x : x is a natural number and multiple of 2, x ≤ 18},

D = {x : x is a natural number < 10}
(c) E = {p, q, r, s, t}, F = {p, s, t, v}
2. If A = {6, 7, 8, 9, 10}, B = {2, 4, 6, 8} and C = {3, 6, 9, 12}, prove that:
(a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

3. If X = {x : x is an even natural number}, Y = {x : x is an odd natural number},
Z = {x : x is a natural number} and V = {x : x is a prime number}, find the following sets.
(a) X ∩ Y (b) Y ∩ V (c) Y ∩ Z

(d) X ∩ V (e) Z ∩ V (f) (X ∩ Y) ∪ (Z ∩ V)
4. If K = {5, 10, 15, 20, 25}, L = {10, 20, 30, 40}, M = {4, 8, 12, 16, 20, 24} and
P = {8, 10, 12, 14, 16, 20, 24} then find:
(a) K – L (b) M – P (c) P – K
(d) L – M (e) K – P (f) M – L

5. If A and B are two sets such that n(A ∪ B) = 18, n(A) = 8 and n(B) = 15, find n(A ∩ B).
6. In a class of 35 students, 24 students like to eat apples and 16 students like to eat oranges. Also,
each student likes to eat at least one of the two fruits. How many students like to eat both the fruits?

AT A GLANCE

¾ A set is a collection of well-defined and distinct objects.

¾ A set is known as a finite set if it has a finite number of elements.
¾ A set is known as an infinite set if it has an infinite number of elements.
¾ A set which contains no element is called an empty set.
¾ A set which contains only one element is called a singleton set.
¾ Two sets are said to be equal if they have the same elements.

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