# Class 11 RD Sharma Solutions – Chapter 1 Sets – Exercise 1.3

**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}.**

**(iv) {x: x is a natural number, x < 8 and simultaneously x > 12}.**

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

**Solution:**

(i)All the numbers ending with 0. Except 0 is divisible by 5 and are even natural number.Therefore, it is not an example of empty set.

(ii)2 is a prime number and is even, and it is the only prime which is even.Therefore, this not an example of the empty set.

(iii)x^{2}– 2 = 0, x^{2}= 2, x = ± √2 ∈ N. There is not natural number whose square is 2.Therefore, it is an example of empty set.

(iv)There is no natural number less than 8 and greater than 12.Therefore, it is an example of the empty set.

(v)No two parallel lines intersect at each other.Therefore, it is an example of empty set.

**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 ∈ N: x > 5}**

**(iv) {x ∈ N: x < 200}**

**(v) {x ∈ Z: x < 5}**

**(vi) {x ∈ R: 0 < x < 1}.**

**Solution:**

(i)Infinite concentric circles can be drawn in a plane.Therefore, it is an infinite set.

(ii)There are just 26 letters in English Alphabets.Therefore, it is finite set.

(iii)It is an infinite set because, natural numbers greater than 5 is infinite.

(iv)It is a finite set. Since, natural numbers start from 1 and there are 199 numbers less than 200.Therefore, it is a finite set.

(v)It is an infinite set. Because integers less than 5 are infinite.Therefore, it is an infinite set.

(vi)It is an infinite set. Because between two real numbers, there are infinite real numbers.

**Question 3. Which of the following sets are equal?**

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

**(ii) B = {x ∈ R: x ^{2}–2x+1=0}**

**(iii) C = (1, 2, 2, 3}**

**(iv) D = {x ∈ R : x ^{3} – 6x^{2}+11x – 6 = 0}**

**Solution:**

A set is said to be equal with another set if all elements of both the sets are equal and same.

A = {1, 2, 3}

B ={x ∈ R: x

^{2 }– 2x+1=0}x

^{2 }– 2x+1 = 0(x–1)

^{2}= 0Therefore, x = 1.

B = {1}

C= {1, 2, 2, 3}

In sets we do not repeat elements hence C can be written as {1, 2, 3}

D = {x ∈ R: x

^{3}– 6x^{2}+11x – 6 = 0}For x = 1, x

^{2}–2x+1=0= (1)

^{3}–6(1)^{2}+11(1)–6= 1–6+11–6

= 0

For x =2,

= (2)

^{3}–6(2)^{2}+11(2)–6= 8–24+22–6

= 0

For x =3,

= (3)

^{3 }– 6(3)^{2}+11(3)–6= 27–54+33–6

= 0

Therefore, D = {1, 2, 3}

Hence, the set A, C and D are equal.

**Question 4. Are the following sets equal?**

**A={x: x is a letter in the word reap},**

**B={x: x is a letter in the word paper},**

**C={x: x is a letter in the word rope}.**

**Solution:**

For A

Letters in word reap

A ={R, E, A, P} = {A, E, P, R}

For B

Letters in word paper

B = {P, A, E, R} = {A, E, P, R}

For C

Letters in word rope

C = {R, O, P, E} = {E, O, P, R}.

Set A = Set B

Because every element of set A is present in set B

But Set C is not equal to either of them because all elements are not present.

**Question 5. From the sets given below, pair the equivalent sets:**

**A= {1, 2, 3}, B = {t, p, q, r, s}, C = {α, β, γ}, D = {a, e, i, o, u}.**

**Solution:**

Equivalent set are different from equal sets, Equivalent sets are those which have equal number of elements they do not have to be same.

A = {1, 2, 3}

Number of elements = 3

B = {t, p, q, r, s}

Number of elements = 5

C = {α, β, γ}

Number of elements = 3

D = {a, e, i, o, u}

Number of elements = 5

Therefore, Set A is equivalent with Set C.

Set B is equivalent with Set D.

### 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 letter of word “FOLLOW”}**

**Solution:**

(i)A = {2, 3}B = x

^{2}+ 5x + 6 = 0x

^{2}+ 3x + 2x + 6 = 0x(x+3) + 2(x+3) = 0

(x+3) (x+2) = 0

x = -2 and -3

= {–2, –3}

Since, A and B do not have exactly same elements hence they are not equal.

(ii)Every letter in WOLFA = {W, O, L, F} = {F, L, O, W}

Every letter in FOLLOW

B = {F, O, L, W} = {F, L, O, W}

Therefore, A and B have same number of elements which are exactly same, hence they are equal sets.

**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}**

**Solution:**

A = {0, a}

B = {1, 2, 3, 4}

C = {4, 8, 12}

D = {3, 1, 2, 4} = {1, 2, 3, 4}

E = {1, 0}

F = {8, 4, 12} = {4, 8, 12}

G = {1, 5, 7, 11}

H = {a, b}

Equivalent sets:

i. A, E, H (They have exactly two elements in them)

ii. B, D, G (They have exactly four elements in them)

iii. C, F (They have exactly three elements in them)

Equal sets:

i. B, D (all of them have exactly the same elements, Hence they are equal)

ii. C, F (all of them have exactly the same elements, Hence they are equal)

**Question 8. Which of the following sets are equal?**

**A = {x : x ∈ N, x < 3}**

**B = {1, 2}, C= {3, 1}**

**D = {x : x ∈ N, x is odd, x < 5}**

**E = {1, 2, 1, 1}**

**F = {1, 1, 3}**

**Solution:**

A = {1, 2}

B = {1, 2}

C = {3, 1}

D = {1, 3} (since, the odd natural numbers less than 5 are 1 and 3)

E = {1, 2} (since, repetition is not allowed)

F = {1, 3} (since, repetition is not allowed)

Therefore, Sets A, B and E are equal.

C, D and F are equal.

**Question 9. Show that the set of letters needed to spell “CATARACT” and the set of letters needed to spell “TRACT” are equal.**

**Solution:**

For “CATARACT”

Distinct letters are

{C, A, T, R} = {A, C, R, T}

For “TRACT”

Distinct letters are

{T, R, A, C} = {A, C, R, T}

As it is seen that the letters need to spell cataract is equal to set of letters need to spell tract.

Therefore, the two sets are equal.

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