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## Tamilnadu Samacheer Kalvi 9th Maths Solutions Chapter 1 Set Language Ex 1.2

Question 1.

Find the cardinal number of the following sets.

(i) M = {p, q, r, s, t, u}

(ii) P = {x : x = 3n + 2, n ∈ W and x < 15}

(iii) Q = {y : y = \(\frac{4}{3n}\), n ∈ N and 2 < n ≤ 5}

(iv) R = {x : x is an integer, x ∈ Z and – 5 ≤ x < 5}

(v) S = The set of all leap years between 1882 and 1906.

Solution:

(i) n (M) = 6

(ii) n (P) = 5 [n = {0, 1, 2, 3 . . . . 14}]

(iii) Since n = {3, 4, 5} ; n (Q) = 3

(iv) X = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4} ∴ n (R) = 10

(v) S = {1884, 1888, 1892, 1896, 1904}; n (S) = 5

Question 2.

Identify the following sets as finite or infinite.

(i) X = The set of all districts in Tamilnadu.

(ii) Y = The set of all straight lines passing through a point.

(iii) A = {x : x ∈ Z and x < 5}

(iv) B = {x : x² – 5x + 6 = 0, x ∈ N}

Solution:

(i) Finite

(ii) Infinite set (many lines can be drawn from a point)

(iii) Infinite set {A = ……. -2, -1, 0, 1, 2, 3, 4}

(iv) Finite set [x² – 5x + 6 = 0 ⇒ (x – 3) (x – 2) = 0; x = 3 and 2]

Question 3.

Which of the following sets are equivalent or unequal or equal sets?

(i) A = The set of vowels in the English alphabets.

B = The set of all letters in the word “VOWEL”

(ii) C = {2, 3, 4, 5}

D = {x : x ∈ W, 1 < x < 5}

(iii) X = {x : x is a letter in the word “LIFE”}

Y = {F, I, L, E}

(iv) G = {x : x is a prime number and 3 < x < 23}

H = {x : x is a divisor of 18}

Solution:

(i) Equivalent set [n(A) = n(B) = 5] ∴ A ≈ B

(ii) Unequal sets [C = {2, 3, 4, 5}; D = {2, 3, 4}]

(iii) Equal sets [X = {L, I, F, E}; Y = {F, I, L, E} [n(X) = 4 = n(Y)] ∴ X ≈ Y

(iv) Equivalent sets [G = {5, 7, 11, 13, 17, 19}; H = {1, 2, 3, 6, 9, 18}]

[n(G) = n(H) = 6 ∴ G ≈ H)]

Question 4.

Identify the following sets as null set or singleton set.

(i) A = {x : x ∈ N, 1 < x < 2}

(ii) B = The set of all even natural numbers which are not divisible by 2

(iii) C = {0}.

(iv) D = The set of all triangles having four sides.

Solution:

(i) Null set [No natural numbers is in between 1 and 2]

(ii) Null set [All the even natural numbers are not divisible by 2]

(Hi) Singleton set [n (C) = 1]

(iv) Null set [All the triangles has 3 sides]

Question 5.

State which pairs of sets are disjoint or overlapping?

(i) A = {f, i, a, s} and B = {a, n, f, h, s}

A = {f, i, a, s} and B = {a, n, f, h, s}

A and B are overlapping sets

(ii) C = {x : x is a prime number, x > 2} and D = {x : x is an even prime number}

C= {3, 5, 7…….}

D = {2}

C and D are disjoint sets

(iii) E = {x : x is a factor of 24} and F = {x : x is a multiple of 3, x < 30}

E = {1, 2, 3, 4, 6, 8, 12, 24}

F = {3, 6, 9, 12, 15, 18, 21, 24, 27} [Hint: E ∩ F = {3, 6, 24, …….}]

E and F are overlapping sets

Question 6.

If S = {square, rectangle, circle, rhombus, triangle}. List the elements of the following subset of S.

(i) The set of shapes which have 4 equal sides.

(ii) The set of shapes which have radius.

(iii) The set of shapes in which the sum of all interior angles is 180°.

(iv) The set of shapes which have 5 sides.

Solution:

(i) Subset of S = {square, rhombus}

(ii) Subset of S = {circle}

(iii) Subset of S = {triangle}

(iv) Subset of S = { }

Question 7.

If A = {a,{a, b}}, write all the subsets of A.

Solution:

A = {a, {a, b}}

Subset of A are Ø, {a}, {a, b}, {a, {a, b}} (or) { }, {a}, {a,b, {a,{a,b}}

P(A) = {Ø, {a}, {a, b}, {a {a, b}} (or) {{ }, {a}, {a,b, {a,{a,b}}

Question 8.

Write down the power set of the following sets.

(i) A = {a, b}

(ii) B = {1, 2, 3}

(iii) D = {p, q, r, s}

(iv) E = Ø

Solution:

(i) A = {a, b)

P(A) = {{},{a},{b}, {a, b}}

(ii) B = {1, 2, 3}

P(B) = {{}, {1}, {2}, {3}, {1,2}, {2, 3}, {1,3}, {1,2,3}}

(iii) D = {p, q, r, s}

P(D) = {{},{p},{q},{r},{s},{p, q} {p, r} {p, s}

{q, r}, {q, s}, {r, s}, {p, q, r} {q, r, s}

{p, r, s} {p, q, s} {p, q, r, s}}

(iv) E = Ø

P(E) = {{}}

Note: (empty set is the subset of all the sets)

Question 9.

Find the number of subsets and the number of proper subsets of the following sets.

(i) W = {red, blue, yellow}

(ii) X = {x² : x ∈ N, x² ≤ 100}

Solution:

(i) W = {red, blue, yellow}

n (W) = 3

The number of subsets of W = n [P(W)] = 2^{m}

= 2^{3} = 8

Number of proper subsets of W = n[P(W)] – 1

= 8 – 1

= 7 (or)

Number of proper subsets of W = 2^{m} – 1

= 2^{3} – 1 = 8 – 1 = 7

(ii) X = {x^{2} : x ∈ N, x^{2} ≤ 100}.

X= {1,2, 3, 4, …. 10}

n(X) = 10

The number of subsets of X = n[P(X)]

= 2^{m}

= 2^{10} = 1024

Number of proper subsets of X = 2^{m} – 1

= 1024 – 1

= 1023

Question 10.

(i) If n(A) = 4, find n[P(A)]

(ii) If n(A) = 0, find n[P(A)]

(Hi) If n[P(A)] = 256, find n(A)

Solution:

(i) n (A) = 4

n [P(A)] = 2^{m} = 2^{4}

= 16

(ii) n (A) = 0

n [P(A)] = 2^{m} = 2° = 1

(iii) n [P(A)] = 256

2^{m} = 2^{8}

∴ n (A) = 8