## Solutions to set-II

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In this page, 'Solutions to set-II' we are discussing how to do the problems given in problems on set-II.

### Cardinal number

1. Find the cardinal number of sets.

(i) A = {x: x=5ⁿ, n∈ℕ and n< 5}

Solution: A = {5, 10, 15, 20}

n(A) = 4.

(ii) B = {x: x is a consonant in English Alphabet}

Solution: B = {b,c,d,f,g,h,j,k,l,m,n,p,q,r,s,t,v,w,x,y,z}

n(B) = 21.

(iii) C = {x: x is an even prime number}

Solution: C = {2}

n(C) = 1.

(iv) D = {x: x<0,x∈W}

Solution: D = ∅

n(D) = 0

(v) E = {x:-3 ≤ x ≤ 5, x∈ℤ}

Solution: E = {-3,-2,-1,0,1,2,3,4,5}

n(E) = 9.

Finite-Infinite sets

2. Identify the following as finite or infinite sets.

(i) A= {4,5,6,...}

Solution: Infinite

(ii) B = {0,1,2,3,4....75}

Solution: Finite

(iii) C ={x: x is an even natural number}

Solution: Infinite

(iv) D = {x: x is a multiple of 6 and x >0}

Solution: Infinite

(v) E = The set of letters in the word, 'ASTRONOMY'.

Solution: Finite

3.Which of the following sets are equivalent?

(i) A = {2,4,6,8,10}, B= {1,3,5,7,9}

Solution: Since both A and B are having equal number of elements, but not same elements, they are **equivalent.**

(ii) X = {x:x∈ℕ, 1<x<6}, Y={x: x is a vowel in the English Alphabet}

Solution: n(X) =4 and n(Y) =5, so they are **not equivalent**.

(iii) P = { x: xis a prime number and 5 < x < 23}

Q = {x: x∈W, 0 ≤ x < 5}

Solution: n(P) =5 and n(Q) =5 so they are **equivalent**.

4. Which of the following sets are equal?

(i) A= {1,2,3,4}, B= {4,3,2,1}

Solution: Equal.

(ii) A= {4,8,12,16}, B = {8,4,16,18}

Solution: Not equal.

(iii) X= {2,4,6,8}

Y = {x: x is a positive even integer 0 < x < 10}

Solution: Equal [Since X = {2,4,6,8}, Y = {2,4,6,8}]

(iv) P = {x: x is a multiple of 10, x∈ℕ}

Q= {10,15,20,25,30....}

Solution: Not equal.

5. From the sets given below, select equal sets.

A= {12,14,18,22}, B={11,12,13,14}, C= {14, 18,22,24}

D= {13,11,12,14}, E = {-11, 11}, F= {10, 19}, G= {11, -11} and H= {10, 11}

Solution: B and D are equal sets.

E and G are equal sets.

6. Is ∅ = {∅}? why?

Solution: They are not equal sets. ∅ is an empty set, but {∅} contains one element.

7. Which of the sets are equal sets? State the reason.

0, ∅, {0}, {∅}.

Solution: None of the sets are equal sets. Because 0 is an integer, not a set. ∅ is an empty set. {0}, {∅} are singleton sets but having different elements.

Students can solve the problems on their own, compare the answer with the solutions discussed above in 'Solutions to set-II'. If you are having any doubt you can contact us through mail, we will help you to clear your doubts.

Set theory

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