Question 1: Which of the following are examples of empty set?

(i) Set of all even natural numbers divisible by 5 .

(ii) Set of all even prime numbers

(iii) \{x : x^2 -2 = 0 and x is rational \}

(vi) \{x : x is a natural number , x < 8 and simultaneously x > 12\}

(v) \{x: x is a point common to any two parallel lines \}

Answer:

(i) Set of all even natural numbers divisible by 5 would mean a set \{  x : x = 2 (5n), n \in N, n >0 \} or \{ 10, 20, 30, 40, \cdots \} . This has elements and therefore not an empty set.

(ii) Set of all even prime numbers would me a set \{ 2  \}  . This has elements and therefore not an empty set.

(iii) \{x : x^2 -2 = 0 and x is rational \} . This would mean that x = \pm \sqrt{2} . \sqrt{2} is irrational number. Hence this is an empty set.

(vi) \{x : x is a natural number , x < 8 and simultaneously x > 12\} . There are no number which are less than 8 and also greater than 12 simultaneously. Hence this is an empty set.

(v) \{x: x is a point common to any two parallel lines \} . This is an empty set because two parallel lines never intersect.

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Question 2: Which of the following sets are finite and which are infinite?

(i) Set of concentric circles in a plane.

(ii) Set of letters of the English Alphabets.

(iii) \{x \in N : x > 5\}

(iv) \{x \in N : x <200\}

(v) \{x \in Z : x < 5\}

(vi) \{x \in R : 0 < x < 1\}

Answer:

(i) Set of concentric circles in a plane: There can be infinite concentric circles. Therefore this is an INFINITE set.

(ii) Set of letters of the English Alphabets: There are only 26 English alphabets \{A, B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U, \\ V,W,X,Y,Z \} . Hence this is an FINITE set.

(iii) \{x \in N : x > 5\} . This would be a set \{ 6, 7, 8, \cdots \} . There will be infinite elements in the set. Therefore this is an INFINITE set.

(iv) \{x \in N : x <200\} . This set will contain \{ 1, 2, 3, 4, \cdots , 198, 199 \} . Hence this is an FINITE set.

(v) \{x \in Z : x < 5\} . This set would be \{ \cdots, -4, -3, -2, -1, 0, 1,2, 3, 4 \} . There will be infinite elements in the set. Therefore this is an INFINITE set.

(vi) \{x \in R : 0 < x < 1\} . This set would  include all decimals, rational numbers and irrational numbers between 0 and 1 . There will be infinite elements in the set. Therefore this is an INFINITE set.

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Question 3: Which of the following sets are equal?

(i) A = \{1, 2, 3\}

(ii) B = \{ x \in R : x^2 -2x + 1 = 0\}

(iii) C = \{1, 2, 2, 3\}

(vi) D = \{x \in R : x^3 - 6x^2 + 11x -6=0\}

Answer:

Note: EQUAL SETS – Two sets A and B are said to be equal if every element of A is a member of B , and every element of B is a member of A .

Set B = \{ x \in R : x^2 -2x + 1 = 0 \} is \{ 1 \}

x^2 -2x + 1 = 0 \Rightarrow (x-1)^2 = 0 \Rightarrow x = 1

C = \{1, 2, 2, 3\} this can also be written as C = \{1, 2, 3\} as repeating element can be eliminated.

D = \{x \in R : x^3 - 6x^2 + 11x -6=0\} is \{1, 2, 3, \}

Hence A = C = D

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Question 4: Are the following sets equal?

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

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

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

Answer:

A = \{x : x is a letter in the word reap \} would mean \{ r, e, a, p \}

B = \{x : x is a letter in the word paper \} would mean \{ p, a, e, r  \}

C = \{x : x is a letter in the word rope \} would mean \{ r, o, p, e \}

Therefore none of them are equal.

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Question 5: From the sets given below, find the pair the equivalent sets: A = \{1, 2, 3\} , B = \{t, p, q, r, s\} , C = \{\alpha , \beta , \gamma \} , D = \{a, e, i, o ,u\}

Answer:

EQUIVALENT SETS: Two finite sets A and B are equivalent if their cardinal numbers are same. i.e. n(A) = n(B) .

Therefore A = \{1, 2, 3\} \Rightarrow n(A) = 3

B = \{t, p, q, r, s\} \Rightarrow n(B) = 5

C = \{\alpha , \beta , \gamma \} \Rightarrow n(C) = 3

D = \{a, e, i, o ,u\} \Rightarrow n(D) = 5

'A \ and \ C'   and 'B \ and \ C' are equivalent sets.

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Question 6: Are the following pairs of sets equal? Give reasons

(i) A = \{2, 3\} , B = \{x : x is a solution of x^2 + 5x + 6 = 0\}

(ii) A = \{x : x is a letter of the word 'WOLF' \} , B = \{x : x is a letter of the word 'FOLLOW' \}

Answer:

(i) A = \{2, 3\}

B = \{x : x is a solution of x^2 + 5x + 6 = 0\} \Rightarrow \{ -2, -3 \}

Therefore NOT EQUAL.

(ii) A = \{x : x is a letter of the word 'WOLF' \} \Rightarrow \{ W, O, L, F \}

B = \{x : x is a letter of the word 'FOLLOW' \} \Rightarrow \{ F, O, L, W \} \ or \  \{ W, O, L, F \}

Hence A and B are EQUAL.

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Question 7: From the sets given below, select equal sets and equivalent sets

A = \{0, a\} B = \{1, 2, 3, 4\}, C = \{4, 8, 12\}, D = \{3, 1, 2, 4\}, E = \{1, 0\}, F = \{8, 4, 12\}, G = \{1, 5, 7, 11\} , H = \{a, b\}

Answer:

A = \{0, a\} \Rightarrow n(A) = 2           B = \{1, 2, 3, 4\} \Rightarrow n(B) = 4

C = \{4, 8, 12\} \Rightarrow n(C) = 3           D = \{3, 1, 2, 4\} \Rightarrow n(D) = 4

E = \{1, 0\} \Rightarrow n(E) = 2           F = \{8, 4, 12\} \Rightarrow n(F) = 3

G = \{1, 5, 7, 11\} \Rightarrow n(G) = 4           H = \{a, b\} \Rightarrow n(H) = 2

Hence we can see that B and D are equal. Also C and F are equal sets.

A, E and H are equivalent. Also C and F are equivalent sets. Also D and G are equivalent.

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Question 8: Which of the following sets are equal?

A = \{ x: x \in N, x < 3\}, \ \ \ \ \ B = \{1, 2\}, C = \{3, 1\}, D = \{x : x \in N, x \ is \ odd \ , x < 5\}, \ \ \ E = \{1, 2, 1,1\}, \ \ \ F  = \{1, 1, 3\}

Answer:

A = \{ x: x \in N, x < 3\} \ or \ A = \{ 1, 2 \} \Rightarrow n(A) = 2

B = \{1, 2\} \Rightarrow n(B) = 2

C = \{3, 1\} \Rightarrow n(C) = 2

D = \{x : x \in N, x \ is \ odd \ , x < 5\} \ or \ D = \{1, 3 \} \Rightarrow n(D) = 2

E = \{1, 2, 1,1\} \Rightarrow n(E) = 2

F  = \{1, 1, 3\} \Rightarrow n(F) = 2

Therefore A = B = E and C = D + F

They are also all equivalent sets as they all have 2 elements each.

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Question 9: Show that the set of letters needed to spell "CATARACT" and the set of letters needed to spell "TRACT" are equal.

Answer:

 Set of letters needed to spell "CATARACT"  = \{ C, A, T, R \}

Set of letters needed to spell "TRACT" = \{ T, R, A, C \}

Therefore they are equal sets as they have the same elements. They are also equivalent sets.

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