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

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

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

Solution:

(i) n(M) = 6

(ii) W = {0, 1, 2, 3, ……. }

if n = 0, x = 3(0) + 2 = 2

if n = 1, x = 3(1) + 2 = 5

if n = 2, x = 3(2) + 2 = 8

if n = 3, x = 3(3)+ 2 =11

if n = 4, x = 3(4) + 2=14

∴ P= {2, 5, 8, 11, 14}

n(P) = 5

(iii) N = {1,2, 3, 4, …..}

n ∈ {3, 4, 5}

n(Q) = 3

(iv) x ∈ z

R = {-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 : x^{2} – 5x + 6 = 0, x ∈ N}

Solution:

(i) Finite set

(ii) Infinite set

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

∴ Infinite set

(iv) x^{2} – 5x + 6 = 0

(x – 3) (x – 2) = 0

B = {3, 2}

∴ Finite set.

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 = A = { 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) A = {a, e, i, o, u}

B = {V, O,W, E, L}

The sets A and B contain the same number of elements.

∴ Equivalent sets

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

D = {2, 3, 4}

∴ Unequal sets

(iii) X = {L, I, F, E}

Y = {F, I, L, E}

The sets X and Y contain the exactly the same elements.

∴ Equal sets.

(iv) G = {5, 7, 11, 13, 17, 19}

H = {1, 2, 3, 6, 9, 18}

∴ Equivalent sets.

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) A = { } ∵ There is no element in between 1 and 2 in Natural numbers.

∴ Null set

(ii) B = { } ∵ All even natural numbers are divisible by 2.

∴ B is Null set

(iii) C = {0}

∴ Singleton set

(iv) D = { }

∵ No triangle has four sides.

∴ D is a Null set.

Question 5.

State which pairs of sets are disjoint or overlapping?

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

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

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

Solution:

(i) A = {f, i, a, s}

B = {a, n, f, h, s}

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

Since A ∩ B ≠ ϕ , A and B are overlapping sets.

(ii) C = {3, 5, 7, 11, ……}

D = {2}

C ∩ D = {3, 5, 7, 11, …… } ∩ {2} = { }

Since C ∩ D = Ø, C and D are disjoint sets.

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

F = {3, 6, 9, 12, 15, 18, 21, 24, 27}

E ∩ F = {1, 2, 3, 4, 6, 8, 12, 24} ∩ {3, 6, 9, 12, 15, 18, 21, 24, 27}

= {3, 6, 12, 24}

Since E ∩ F ≠ ϕ, 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) {Square, Rhombus}

(ii) {Circle}

(iii) {Triangle}

(iv) Null set.

Question 7.

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

Solution:

A= {a, {a, b}} subsets of A are { } {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) The subsets of A are Ø, {a}, {b}, {a, b}

The power set of A

P(A ) = {Ø, {a}, {b}, {a,b}}

(ii) The subsets of B are ϕ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}

The power set of B

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

(iii) The subset of D are Ø, {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}}

The power set of D

P(D) = {Ø, {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) The power set of E

P(E) = { }.

Question 9.

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

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

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

Solution:

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

Then n(W) = 3

The number of subsets = n[P(W)] = 2^{3} = 8

The number of proper subsets = n[P(W)] – 1 = 2^{3} – 1 = 8 – 1 = 7

(ii) Given X ={1,2,3, }

X^{2} = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}

n(X) = 10

The Number of subsets = n[P(X)] = 2^{10} = 1024

The Number of proper subsets = n[P(X)] – 1 = 2^{10} – 1 = 1024 – 1 = 1023.

Question 10.

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

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

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

Solution:

(i) n( A) = 4

n[ P(A)] = 2n = 2^{4} = 16

(ii) n(A) = 0

n[P(A)] = 2^{0} = 1

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

n[P(A)] = 28

∴ n(A) = 8.