Sets — Exercise 7(B)

Write the following set in Roster form and write its cardinal number:

A = {x : x ∈ W, x < 6}

Answer

The whole numbers less than 6 are 0, 1, 2, 3, 4 and 5.

The number of distinct elements in a finite set A is called its cardinal number, denoted by n(A).

Here the number of distinct elements = 6

Hence, A = {0, 1, 2, 3, 4, 5} and n(A) = 6.

Write the following set in Roster form and write its cardinal number:

B = {x : x ∈ N is a multiple of 4, x < 30}

Answer

The multiples of 4 that are less than 30 are 4, 8, 12, 16, 20, 24 and 28.

The number of distinct elements in a finite set B is called its cardinal number, denoted by n(B).

Here the number of distinct elements = 7

Hence, B = {4, 8, 12, 16, 20, 24, 28} and n(B) = 7.

Write the following set in Roster form and write its cardinal number:

C = {x : x ∈ N, x is a perfect square, x < 30}

Answer

The perfect squares less than 30 are 1, 4, 9, 16 and 25.

The number of distinct elements in a finite set C is called its cardinal number, denoted by n(C).

Here the number of distinct elements = 5

Hence, C = {1, 4, 9, 16, 25} and n(C) = 5.

Write the following set in Roster form and write its cardinal number:

D = {x : x ∈ N, x is a composite number, 10 < x < 18}

Answer

The composite numbers x such that 10 < x < 18 are 12, 14, 15 and 16.

The number of distinct elements in a finite set D is called its cardinal number, denoted by n(D).

Here the number of distinct elements = 4

Hence, D = {12, 14, 15, 16} and n(D) = 4.

Write the following set in Roster form and write its cardinal number:

E = {x : x is an odd prime number, x < 18}

Answer

The odd prime numbers less than 18 are 3, 5, 7, 11, 13 and 17.

The number of distinct elements in a finite set E is called its cardinal number, denoted by n(E).

Here the number of distinct elements = 6

Hence, E = {3, 5, 7, 11, 13, 17} and n(E) = 6.

Write the following set in Roster form and write its cardinal number:

F = {x : x ∈ N, 5 < x < 6}

Answer

There is no natural number lying between 5 and 6. So, F is an empty set.

The number of distinct elements in a finite set F is called its cardinal number, denoted by n(F).

Here the number of distinct elements = 0

Hence, F = { } and n(F) = 0.

Write the following set in Roster form and write its cardinal number:

G = {x : x is a letter in the word 'ALLAHABAD'}.

Answer

The letters in the word 'ALLAHABAD' are A, L, L, A, H, A, B, A, D. Writing each letter only once, we get A, L, H, B, D.

The number of distinct elements in a finite set G is called its cardinal number, denoted by n(G).

Here the number of distinct elements = 5

Hence, G = {A, L, H, B, D} and n(G) = 5.

Let :

A = {x : x is a letter in the word 'INCREASE'}

B = {x : x is a vowel in the word 'INCREASE'}

and C = {x : x is a consonant in the word 'INCREASE'}

(i) Write A, B and C in Roster form.

(ii) Find n(A), n(B) and n(C).

Answer

(i) The letters in the word 'INCREASE' are I, N, C, R, E, A, S, E. Writing each letter only once, we get I, N, C, R, E, A, S.

So, A = {I, N, C, R, E, A, S}.

The vowels among these letters are I, E, A.

So, B = {I, E, A}.

The consonants among these letters are N, C, R, S.

So, C = {N, C, R, S}.

Hence, A = {I, N, C, R, E, A, S}, B = {I, E, A} and C = {N, C, R, S}.

(ii) A has 7 elements, B has 3 elements and C has 4 elements.

Hence, n(A) = 7, n(B) = 3 and n(C) = 4.

Which of the following sets are equal?

(i) A = {x : x ∈ N, 1 < x < 3} and B = {x : x is prime, x is even}.

(ii) C = {x : x ∈ W, x < 8} and D = {x : x ∈ N, x < 9}.

(iii) E = {x : x ∈ N, x is odd, x < 6} and F = {x : x ∈ N, x is a factor of 15, x < 15}.

(iv) G = {x : x is a prime factor of 12} and H = {x : x is prime, x < 4}.

(v) φ and {0}

Answer

(i) A = {x : x ∈ N, 1 < x < 3}. The only natural number lying between 1 and 3 is 2. So, A = {2}.

B = {x : x is prime, x is even}. The only even prime number is 2. So, B = {2}.

The elements of A and B are exactly the same.

Hence, A = B.

(ii) C = {x : x ∈ W, x < 8} = {0, 1, 2, 3, 4, 5, 6, 7}.

D = {x : x ∈ N, x < 9} = {1, 2, 3, 4, 5, 6, 7, 8}.

C contains 0 but not 8, while D contains 8 but not 0. So the elements are not the same.

Hence, C ≠ D.

(iii) E = {x : x ∈ N, x is odd, x < 6} = {1, 3, 5}.

F = {x : x ∈ N, x is a factor of 15, x < 15}. The factors of 15 are 1, 3, 5 and 15. Those less than 15 are 1, 3 and 5. So, F = {1, 3, 5}.

The elements of E and F are exactly the same.

Hence, E = F.

(iv) G = {x : x is a prime factor of 12}. Since 12 = 2² × 3, the prime factors of 12 are 2 and 3. So, G = {2, 3}.

H = {x : x is prime, x < 4}. The prime numbers less than 4 are 2 and 3. So, H = {2, 3}.

The elements of G and H are exactly the same.

Hence, G = H.

(v) φ is the empty set, containing no element at all. But {0} is a singleton set, containing one element, namely 0.

Hence, φ ≠ {0}.

Which of the following are empty sets?

(i) A = {x : x ∈ N, x + 3 = 3}

(ii) B = {x : x ∈ W, x + 5 = 5}

(iii) C = {x : x ∈ N, x is even, x is divisible by 3} [Hint : C = {6, 12, 18, .......}].

(iv) D = {x : x is a prime number, x < 2}

(v) E = {x : x ∈ N, x < 1}

(vi) F = {x : x ∈ N, 2x = 1}

(vii) G = {0}

Answer

(i) Solving,

⇒ x + 3 = 3

⇒ x = 0.

But 0 is not a natural number, so no natural number satisfies the condition.

Hence, A is an empty set.

(ii) Solving,

⇒ x + 5 = 5

⇒ x = 0.

Since 0 is a whole number, B = {0}, which has one element.

Hence, B is not an empty set.

(iii) An even number that is divisible by 3 is a multiple of 6. So, C = {6, 12, 18, ......}, which has elements.

Hence, C is not an empty set.

(iv) The smallest prime number is 2, so there is no prime number less than 2.

Hence, D is an empty set.

(v) The smallest natural number is 1, so there is no natural number less than 1.

Hence, E is an empty set.

(vi) Solving,

⇒ 2x = 1

⇒ x = $\dfrac{1}{2}$.

But $\dfrac{1}{2}$ is not a natural number, so no natural number satisfies the condition.

Hence, F is an empty set.

(vii) G = {0} has one element, namely 0.

Hence, G is not an empty set.

Hence, the empty sets are A, D, E and F.

Classify the following into finite and infinite sets:

(i) A = {x : x ∈ N, x < 100}

(ii) B = {x : x ∈ I, x < 100}

(iii) C = {x : x ∈ N, x > 100}

(iv) D = {x : x ∈ I, x > -1}

(v) E = {x : x ∈ N, x is a multiple of 3}

(vi) F = {x : x ∈ N, x is a factor of 1000}

(vii) G = {x : x is a point on an arc of a circle}

(viii) H = {x : x is a diameter of a circle}

Answer

(i) A = {x : x ∈ N, x < 100} = {1, 2, 3, ......, 99}.

The counting of its elements comes to an end.

Hence, A is a finite set.

(ii) B = {x : x ∈ I, x < 100} = {......, -3, -2, -1, 0, 1, ......, 99}.

The listing extends endlessly towards the negative side.

Hence, B is an infinite set.

(iii) C = {x : x ∈ N, x > 100} = {101, 102, 103, ......}.

The listing never comes to an end.

Hence, C is an infinite set.

(iv) D = {x : x ∈ I, x > -1} = {0, 1, 2, 3, ......}.

The listing never comes to an end.

Hence, D is an infinite set.

(v) E = {x : x ∈ N, x is a multiple of 3} = {3, 6, 9, 12, ......}.

The listing never comes to an end.

Hence, E is an infinite set.

(vi) A number has only a limited number of factors. So the factors of 1000 can be counted and the counting comes to an end.

Hence, F is a finite set.

(vii) An arc of a circle contains an unlimited (uncountable) number of points.

Hence, G is an infinite set.

(viii) A circle has an unlimited number of diameters.

Hence, H is an infinite set.

Find whether the given statement is true or false:

(i) If P and Q are two sets such that n(P) = n(Q), then P = Q.

(ii) Two equivalent sets are always equal.

(iii) Two equal sets are always equivalent.

(iv) Let A = {x : x is a letter in the word 'ENGINEER'}. Then, n(A) = 5.

(v) Let B = {x : x is a consonant in the word 'MANAGEMENT'}. Then, n(B) = 4.

(vi) Let C = {x : x ∈ N, x is neither prime nor composite}. Then, n(C) = 1.

(vii) n(φ) = 0.

(viii) {x : x ∈ I, x is neither positive nor negative} = φ.

Answer

(i) Two sets with the same cardinal number are equivalent, but they need not be equal. For example, {1, 2} and {3, 4} have the same cardinal number but are not equal.

False

(ii) Equivalent sets have the same cardinal number, but their elements may be different. So they need not be equal.

False

(iii) Equal sets have exactly the same elements, so they must have the same cardinal number. Hence, they are always equivalent.

True

(iv) The letters in the word 'ENGINEER' are E, N, G, I, N, E, E, R. Writing each letter only once, we get E, N, G, I, R. So, A = {E, N, G, I, R} and n(A) = 5.

True

(v) The letters in the word 'MANAGEMENT' are M, A, N, A, G, E, M, E, N, T. The consonants among these are M, N, G, M, N, T. Writing each letter only once, we get M, N, G, T. So, B = {M, N, G, T} and n(B) = 4.

True

(vi) The only natural number that is neither prime nor composite is 1. So, C = {1} and n(C) = 1.

True

(vii) The empty set has no element at all, so its cardinal number is 0 i.e. n(φ) = 0.

True

(viii) The integer 0 is neither positive nor negative. So the set is {0}, which has one element and is not the empty set.

False

Let A = {2, 4, 6, 8, 10} and B = {x : x ∈ W, x < 5}.

(i) Find n(A) and n(B).

(ii) Are A and B equivalent sets?

Answer

A = {2, 4, 6, 8, 10}

B = {x : x ∈ W, x < 5} = {0, 1, 2, 3, 4}

(i) A has 5 elements and B has 5 elements.

Hence, n(A) = 5 and n(B) = 5.

(ii) Since n(A) = n(B) = 5, the two sets have the same cardinal number.

Hence, A and B are equivalent sets.