Exercise 3

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1. Which of the following sets are equal?

(i) A = {x : x is a letter in the word FOLLOW}

(ii) B = {x : x is a letter in the word FLOW}

(iii) C = {x : x is a letter in the word WOLF}

(i) A = {x : x is a letter in the word FOLLOW}={F,O,L,W}

(ii) B = {x : x is a letter in the word FLOW}={F,L,O,W}

(iii) C = {x : x is a letter in the word WOLF} ={W,O,L,F}

All elements in sets A , B and C are equal so A,B and C are equal sets (i.e.A=B=C).

2. Consider the following sets and fill up the blank in the statement given below with = or ≠ so as to make the statement true.

A = {1, 2, 3}; B = {The first three natural numbers}

C = {a, b, c, d}; D = {d, c, a, b}

E = {a, e, i, o, u}; F = {set of vowels in English Alphabet}

(i) A .... B (ii) A .... E (iii) C .... D

(iv) D .... F (v) F .... A (vi) D .... E

(vii) F .... B

B = {The first three natural numbers}= {1, 2, 3}

F = {set of vowels in English Alphabet}={a, e, i, o, u}

(i) A = B (ii) A ≠ E (iii) C = D

(iv) D ≠ F (v) F ≠ A (vi) D ≠ E

(vii) F≠ B

3. In each of the following, state whether A = B or not.

(i) A = {a, b, c, d} B = {d, c, a, b} (ii) A = {4, 8, 12, 16} B = {8, 4, 16, 18} (iii) A = {2, 4, 6, 8, 10} B = {x : x is a positive even integer and x ≤ 10} (iv) A = {x : x is a multiple of 10} B = {10, 15, 20, 25, 30, …}

(i) All elements in set A are in set B and all elements in set B are in set A. `therefore` A= B

All elements in set A are not present in set B. `therefore`A ≠B

(iii)B={2,4,6,8,10}

All elements in set A are in set B and all elements in set B are in set A.`therefore` A= B

(iv) A= {10,20,30,40,----}

All elements in set A are not present in set B. `therefore`A ≠B

4. State the reasons for the following :

(i) {1, 2, 3, …., 10} ≠ {x : x ∈ N and 1 < x < 10}

{x : x ∈ N and 1 < x < 10}={2, 3, 4, ..... 9}

so,{1, 2, 3, ..... 10} ≠ {2, 3, 4, ..... 9}

(ii) {2, 4, 6, 8, 10} ≠ {x : x = 2n+1 and x ∈ N}

x = 2x + 1 means x is odd

(iii) {5, 15, 30, 45} ≠ {x : x is a multiple of 15}

x is multiple of 15. So 5 does not exist

(iv) {2, 3, 5, 7, 9} ≠ {x : x is a prime number}

x is prime number but 9 is not a prime number

5. List all the subsets of the following sets.

(i) B = {p, q} (ii) C = {x, y, z} (iii) D = {a, b, c, d} (iv) E = {1, 4, 9, 16} (v) F = {10, 100, 1000}

(i) Subsets of B = {p, q} are {p}, {q}, {p, q}, 𝜙

(ii) Subsets of C = {x, y, z} are {x}, {y}, {z}, {x, y}, {y, z}, {z, x}, {x, y, z}, 𝜙

(iii) Subsets of D = {a, b, c, d} are {a}, {b}, {c}, {d}, {a, b}, {b, c}, {c, d}, {a, c}, {a, d}, {b, d}, {a, b, c}, {b, c, d}, {a, b, d}, {a, c, d}, {a, b, c, d}, 𝜙

(iv)Subsets of E = {1, 4, 9, 16} are 𝜙, {1}, {4}, {9}, {16}, {1, 4}, {1, 9}, {1, 16}, {4, 9}, {4, 16}, {9, 16}, {1, 4, 9}, {1, 9, 16}, {4, 9, 16}, {1, 4, 16}, {1, 4, 9, 16}

(v) Subsets of F={10, 100, 1000} are 𝜙, {10}, {100}, {1000}, {10, 100}, {100, 1000}, {10, 1000}, {10, 100, 1000}

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❮ Finite sets