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Sets (Ex – 1.2)
Question 1.
Which of the following are examples of the null set?
(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers
(iii) {x: x is a natural number, x ≤ 5 and x > 7}
(iv) {y: y is a point common to any two parallel lines}
Solution.
(i) Set of odd natural numbers divisible by 2 is a null set because odd natural numbers are not divisible by 2.
(ii) Set of even prime numbers is {2} which is not a null set.
(iii) {x: x is a natural number, x < 5 and x >7} is a null set because there is no natural number which satisfies x < 5 and x > 7 simultaneously,
(iv) [y: y is a point common to any two parallel lines) is a null set because two parallel lines
do not have any common point.
Question 2.
Which of the following sets are finite or infinite?
(i) The set of months of a year
(ii) {1,2,3,…}
(iii) {1,2,3, …,99,100}
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99
Solution.
(i) The set of months of a year is finite set because there are 12 months in a year.
(ii) {1, 2, 3, …} is an infinite set because there are infinite elements in the set.
(iii) {1, 2, 3, …, 99, 100) is a finite set because the set contains finite number of elements.
(iv) The set of positive integers greater than 100 is an infinite set because there are infinite
number of positive integers greater than 100.
(v) The set of prime numbers less than 99 is a finite set because the set contains finite number of elements.
Question 3.
State whether each of the following set is finite or infinite:
(i) The set of lines which are parallel to the x-axis
(ii) The set of letters in the English alphabet
(iii) The set of numbers which are multiple of 5
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0,0)
Solution.
(i) The set of lines which are parallel to the x-axis is an infinite set because we can draw infinite number of lines parallel to x-axis.
(ii) The set of letters in the English alphabet is a finite set because there are 26 letters in the English alphabet.
(iii) The set of numbers which are multiple of 5 is an infinite set because there are infinite multiples of 5.
(iv) The set of animals living on the earth is a finite set because the number of animals living on the earth is very large but finite.
(v) The set of circles passing through the origin (0, 0) is an infinite set because we can draw infinite number of circles passing through origin of different radii.
Question 4.
In the following, state whether A = B or not:
(i) A = {a, b, c, d};B = {d, c, b, a}
(ii) A = {4, 8, 12, 16};B = {8, 4, 16, 18}
(iii) A = {2, 4, 6, 8, 10}
B = {x : x is positive even integer and x≤ 10}
(iv) A = {x: x isa multiple of 10}
B = {10, 15, 20, 25, 30,…}
Solution.
(i) A = {a, b, c, d} and B = {d, c, b, a} are equal sets because order of elements does not changes a set.
∴ A = B = [a, b, c, d}.
(ii) A = {4, 8, 12, 16} and B = {8, 4, 16, 18} are not equal sets because 12 ∈ A but 12 ∉ B and 18 ∉ B but 18 ∉ A.
(iii) A = {2, 4, 6, 8,10} and B = {x: x is a positive even integer and x ≤ 10) which can be written in roster form as B = (2, 4, 6, 8, 10) are equal sets.
∴ A = B = {2, 4, 6, 8,10).
(iv) A = {x: x is a multiple of 10) can be written in roster form as A = {10, 20, 30, 40,…….. } and
B – {10, 15, 20, 25, 30, ………..} are not equal sets because 15 ∈ B but 15 ∉ A.
Question 5.
Are the following pair of sets equal ? Give reasons.
(i) A = {2, 3}, B={x: x is solution of x2 + 5x + 6 = 0}
(ii) A = {x: x is a letter in the word FOLLOW}
B = {y: y is a letter in the word WOLF}
Solution.
(i) A = (2, 3} and B = {x: x is a solution of x2 + 5x + 6 = 0}
Now, x2 + 5x + 6 = 0 ⇒ x2 + 3x + 2x + 6 = 0 ⇒ (x + 3)(x + 2) = 0 ⇒ x = -3, -2
∴ B = {-2, -3}
Hence, A and B are not equal sets.
(ii) A = {x : x is a letter in the word FOLLOW} = {F, O, L, W}
B = {y: y is a letter in the word WOLF}
= {W, O, L, F}
Hence, A = B = {F, O, L, W}.
Question 6
From the sets given below, select equal sets:
A = {2, 4, 8, 12),
B = {1, 2, 3, 4},
C = {4, 8, 12, 14},
D ={3,1,4,2},
E ={-1, 1},
F ={0, a},
G ={1, -1},
H ={0, 1}
Solution.
From the given sets, we see that sets B and D have same elements and also sets E and G have same elements.
∴ B = D = {1 ,2, 3, 4} and E = G = {-1, 1}.
