Set Concepts — Exercise 6(B)

Write the cardinal number of each of the following sets :

(i) A = Set of days in a leap year.

(ii) B = Set of numbers on the face of a clock.

(iii) C = {x : x ∈ N and x ≤ 7}

(iv) D = Set of letters in the word "PANIPAT".

(v) E = Set of prime numbers between 5 and 15.

(vi) F = {x : x ∈ Z and −2 < x ≤ 5}

(vii) G = {x : x is a perfect square number, x ∈ N and x ≤ 30}.

Answer

(i) A leap year has 366 days, so set A has 366 elements.

n(A) = 366

(ii) The numbers on the face of a clock are 1, 2, 3, ..., 12, so B has 12 elements.

n(B) = 12

(iii) C = {x : x ∈ N and x ≤ 7} = {1, 2, 3, 4, 5, 6, 7}, so C has 7 elements.

n(C) = 7

(iv) The letters of the word PANIPAT are P, A, N, I, P, A, T. Writing each letter only once, we get P, A, N, I and T, i.e. D = {P, A, N, I, T}.

n(D) = 5

(v) The prime numbers between 5 and 15 are 7, 11 and 13, i.e. E = {7, 11, 13}.

n(E) = 3

(vi) F = {x : x ∈ Z and −2 < x ≤ 5} = {−1, 0, 1, 2, 3, 4, 5}, so F has 7 elements.

n(F) = 7

(vii) The perfect square numbers x ∈ N with x ≤ 30 are 1, 4, 9, 16 and 25, i.e. G = {1, 4, 9, 16, 25}.

n(G) = 5

For each set, given below, state whether it is a finite set, infinite set or the null set :

(i) {natural numbers more than 100}.

(ii) A = {x : x is an integer between 1 and 2}.

(iii) B = {x : x ∈ W; x is less than 100}.

(iv) Set of mountains in the world.

(v) {multiples of 8}.

(vi) {even numbers not divisible by 2}.

(vii) {squares of natural numbers}.

(viii) {coins used in India}.

(ix) C = {x | x is a prime number between 7 and 10}.

(x) Planets of the solar system.

Answer

(i) {natural numbers more than 100} = {101, 102, 103, ...}, which never ends. Infinite set

(ii) A = {x : x is an integer between 1 and 2}. There is no integer between 1 and 2. Null set (φ)

(iii) B = {x : x ∈ W; x is less than 100} = {0, 1, 2, ..., 99}, which has a limited number of elements. Finite set

(iv) Set of mountains in the world has an unlimited number of elements. Infinite set

(v) {multiples of 8} = {8, 16, 24, 32, ...}, which never ends. Infinite set

(vi) {even numbers not divisible by 2}. Every even number is divisible by 2, so there is no such number. Null set (φ)

(vii) {squares of natural numbers} = {1, 4, 9, 16, ...}, which never ends. Infinite set

(viii) {coins used in India} has a limited number of elements. Finite set

(ix) C = {x | x is a prime number between 7 and 10}. The numbers between 7 and 10 are 8 and 9 which are not prime, so there is no prime number between 7 and 10. Null set (φ)

(x) Planets of the solar system are limited in number. Finite set

State, which of the following pairs of sets are disjoint :

(i) {0, 1, 2, 6, 8} and {odd numbers less than 10}.

(ii) {birds} and {trees}

(iii) {x : x is a fan of cricket} and {x : x is a fan of football}.

(iv) A = {natural numbers less than 10} and B = {x : x is a multiple of 5}.

(v) {people living in Calcutta} and {people living in West Bengal}

Answer

Two sets are disjoint if they have no element in common.

(i) {0, 1, 2, 6, 8} and {odd numbers less than 10} = {1, 3, 5, 7, 9}. The element 1 is common to both sets. Not disjoint.

(ii) {birds} and {trees}. A bird is not a tree, so they have no element in common. Disjoint.

(iii) {x : x is a fan of cricket} and {x : x is a fan of football}. A person can be a fan of both cricket and football, so the sets can have common elements. Not disjoint.

(iv) A = {natural numbers less than 10} = {1, 2, 3, ..., 9} and B = {x : x is a multiple of 5} = {5, 10, 15, ...}. The element 5 is common to both sets. Not disjoint.

(v) {people living in Calcutta} and {people living in West Bengal}. Calcutta is a city in West Bengal, so the people living in Calcutta also live in West Bengal. Not disjoint.

Hence, only the pair in (ii) is disjoint.

State whether the given pairs of sets are equal or equivalent :

(i) A = {first four natural numbers} and B = {first four whole numbers}.

(ii) A = Set of letters of the word "FOLLOW" and B = Set of letters of the word "WOLF".

(iii) E = {even natural numbers less than 10} and O = {odd natural numbers less than 9}.

(iv) A = {days of the week starting with letter S} and B = {days of the week starting with letter T}.

(v) M = {multiples of 2 and 3 between 10 and 20} and N = {multiples of 2 and 5 between 10 and 20}.

(vi) P = {prime numbers which divide 70 exactly} and Q = {prime numbers which divide 105 exactly}.

(vii) A = {0², 1², 2², 3², 4²} and B = {16, 9, 4, 1, 0}.

(viii) E = {8, 10, 12, 14, 16} and F = {even natural numbers between 6 and 18}.

(ix) A = {letters of the word SUPERSTITION} and B = {letters of the word JURISDICTION}.

Answer

Two sets are equal if they have exactly the same elements, and equivalent if they have the same number of elements.

(i) A = {first four natural numbers} = {1, 2, 3, 4} and B = {first four whole numbers} = {0, 1, 2, 3}.

n(A) = 4 = n(B), but the elements are different.

Equivalent

(ii) A = {letters of FOLLOW} = {F, O, L, W} and B = {letters of WOLF} = {W, O, L, F}.

Both sets have exactly the same elements.

Equal

(iii) E = {even natural numbers less than 10} = {2, 4, 6, 8} and O = {odd natural numbers less than 9} = {1, 3, 5, 7}.

n(E) = 4 = n(O), but the elements are different.

Equivalent

(iv) A = {days of the week starting with letter S} = {Saturday, Sunday} and B = {days of the week starting with letter T} = {Tuesday, Thursday}.

n(A) = 2 = n(B), but the elements are different.

Equivalent

(v) M = {multiples of 2 and 3 between 10 and 20} = {multiples of 6 between 10 and 20} = {12, 18} and N = {multiples of 2 and 5 between 10 and 20} = {multiples of 10 between 10 and 20} = { }.

n(M) = 2 and n(N) = 0, so the sets are neither equal nor equivalent.

None

(vi) 70 = 2 × 5 × 7, so P = {prime numbers which divide 70 exactly} = {2, 5, 7}.

105 = 3 × 5 × 7, so Q = {prime numbers which divide 105 exactly} = {3, 5, 7}.

n(P) = 3 = n(Q), but the elements are different.

Equivalent

(vii) A = {0², 1², 2², 3², 4²} = {0, 1, 4, 9, 16} and B = {16, 9, 4, 1, 0} = {0, 1, 4, 9, 16}.

Both sets have exactly the same elements.

Equal

(viii) E = {8, 10, 12, 14, 16} and F = {even natural numbers between 6 and 18} = {8, 10, 12, 14, 16}.

Both sets have exactly the same elements.

Equal

(ix) A = {letters of SUPERSTITION} = {S, U, P, E, R, T, I, O, N}, so n(A) = 9.

B = {letters of JURISDICTION} = {J, U, R, I, S, D, C, T, O, N}, so n(B) = 10.

Since n(A) ≠ n(B), the sets are neither equal nor equivalent.

None

Examine which of the following sets are the empty sets :

(i) The set of triangles having three equal sides.

(ii) The set of lions in your class.

(iii) {x : x + 3 = 2 and x ∈ N}

(iv) P = {x : 3x = 0}

Answer

(i) The set of triangles having three equal sides. An equilateral triangle has three equal sides, so such triangles exist. Not an empty set.

(ii) The set of lions in your class. There are no lions in a class. Empty set.

(iii) {x : x + 3 = 2 and x ∈ N}

x + 3 = 2

⇒ x = 2 − 3

⇒ x = −1

Since −1 is not a natural number, there is no such x. Empty set.

(iv) P = {x : 3x = 0}

⇒ 3x = 0

⇒ x = 0

So P = {0}, which has one element. Not an empty set.

Hence, the sets in (ii) and (iii) are the empty sets.

State true or false :

(i) All examples of the empty set are equal.

(ii) All examples of the empty set are equivalent.

(iii) If two sets have the same cardinal number, they are equal sets.

(iv) If n(A) = n(B), then A and B are equivalent sets.

(v) If B = {x : x + 4 = 4}, then B is the empty set.

(vi) The set of all points in a line is a finite set.

(vii) The set of letters in your Mathematics book is an infinite set.

(viii) If M = {1, 2, 4, 6} and N = {x : x is a factor of 12}, then M = N.

(ix) The set of whole numbers greater than 50 is an infinite set.

(x) If A and B are two different infinite sets, then n(A) = n(B).

Answer

(i) All examples of the empty set are equal. There is one and only one empty set, so all its examples are equal. True

(ii) All examples of the empty set are equivalent. Each empty set has cardinal number 0, so they are equivalent. True

(iii) If two sets have the same cardinal number, they are only equivalent, not necessarily equal. False

(iv) If n(A) = n(B), then A and B are equivalent sets. By definition of equivalent sets, this is correct. True

(v) If B = {x : x + 4 = 4}, then B is not the empty set. Here x + 4 = 4 ⇒ x = 0, so B = {0}, which is not empty. False

(vi) The set of all points in a line is not a finite set. A line contains an unlimited number of points, so it is an infinite set. False

(vii) The set of letters in your Mathematics book is not an infinite set. The letters used are from the English alphabet, which are limited (26), so it is a finite set. False

(viii) If M = {1, 2, 4, 6} and N = {x : x is a factor of 12}, here N = {1, 2, 3, 4, 6, 12} ≠ M. False

(ix) The set of whole numbers greater than 50 is an infinite set. It is {51, 52, 53, ...}, which never ends. True

(x) If A and B are two different infinite sets, then n(A) = n(B). Two infinite sets are always equivalent. True

Which of the following represents the null set ?

φ, {0}, 0, { }, {φ}.

Answer

φ — This is the symbol for the empty set, so it represents the null set.

{0} — This is a set with one element 0, so it is not the null set.

0 — This is a number, not a set.

{ } — This is a pair of braces with no element in it, so it represents the null set.

{φ} — This is a set with one element φ, so it is not the null set.

Hence, φ and { } represent the null set.