Question 1: Which of the following collections of objects are sets?

(i)  All the months in a year

Answer: YES. It is a well-defined collection of distinct objects. There are 12 months in a year. So, it is a definite set of elements.

A = \{January, February, March, April, May, June, July,  \\ August, September, October, November, December \}

(ii)  All the rivers flowing in UP

Answer: YES. It is a well-defined collection of distinct objects.

A = \{ Ganga, Yamuna \cdots \}

(iii)  All the planets in our solar system

Answer: YES. It is a well-defined collection of distinct objects.

A= \{ Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune \}

(iv) All the interesting dramas written by Premchand

Answer: NO. It is not a well-defined collection of objects.  Had it been “All dramas written by Premchand” then it could have been a set as that would be well defined.

(v) All the short boys in your class

Answer: NO. It is not a well-defined collection of objects.  Had it been “All boys shorter than 5' 3'' , then it would have become well defined. We need a benchmark to have a well-defined list of objects.

(vi)  All the letters of the English Alphabets which precedes K

Answer: YES. It is a well-defined collection of distinct objects.

A = \{ A, B, C, D, E, F, G, H, I, J \}

(vii) All the pet dogs in Nagpur

Answer: YES. It is a well-defined collection of distinct objects. Technically, all pets in Nagpur should be registered with a Government department. This might be a big list but still a well-defined list of objects.

(viii) All the dishonest shop owners in Noida

Answer: No. Not well defined.

(ix)  All the students in your school with age exceeding 15 years

Answer: YES. It is a well-defined collection of distinct objects. School has a list of students that study there. The set would include all whose age is more than 15 years.

(x) All the girls of Gita’s class who are taller than Gita

Answer: YES. It is a well-defined collection of distinct objects. We know all the girls studying in the class, we know Gita’s height and hence the set would contain distinct girls taller than Gita.

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Question 2: Rewrite the following statements using the set notations:

(i)  p is an element of A

Answer: p \in A

(ii)  q does not belong to set B

Answer: q \notin B

(iii) a and b are members of set C

Answer: a, b \in C

(iv)  B and C are equivalent sets

Answer: B \leftrightarrow C

(v) Cardinal number of set E is 15

Answer: n(E) = 15

Note: The number of distinct elements contained in a finite set A is called the cardinal number of A and is denoted by n(A) .

For example, if A = \{1, 2, 3, 4, 5 \} , then n(A)=5

(vi) A is an empty set and B is a non-empty set

Answer: A = \phi and B \neq \phi

Note: A set consisting of no elements is called an empty set or a null set or a void set. It is denoted by \phi  (called phai). We write f = \{  \}

(vii) 0 is a whole number, but 0 is not a natural number

Answer: 0 \in W but 0 \notin N

Note: Whole numbers (W) : The numbers \{0, 1, 2, 3,  \cdots \} .

Natural numbers (N) : The counting numbers \{ 1, 2, 3,  \cdots \} , are called natural numbers

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Question 3: Describe the following sets in roster form

(i) B = \{x | x \in W, x \leq 6 \}

Answer: B = \{0, 1, 2, 3, 4, 5, 6 \}

(ii) C = \{ x | x \ is \ a \ factor \ of \ 32 \}

Answer: C = \{ 1, 2, 4, 8, 16, 32 \}

Note: You can calculate the factors using a tree method.

(iii) E = \{ x | x = (2n+1), n \in W, n \leq 4 \}

Answer: E = \{ 1, 3, 5, 7, 9 \}

Note: n \in W and n \leq 4 . This means n is 0, 1, 2, 3 , and 4 . Now calculate x based on the given formula. Example: When n = 0, x = 1 .

(iv) F = \{ x | x = n^2, n \in N, 2 \leq n \leq 5 \}

Answer: F = \{ 4, 9, 16, 25 \}

Note: n \in N and 2 \leq n \leq 5 . This means n is 2,3,4,5 . Now substitute the value of n to calculate x .

(v) G = \{ x | x = \frac{n}{n+3} , n \in N \ and \ n \leq 5 \}

Answer: G = \{ 1/4, 2/5, 3/6. 4/7, 5/8 \}

Note: n \in N and n \leq 5 , which means n is 1, 2, 3, 4, 5 . Now substitute the value of n in the formula given for x .

(vi) H = \{ x | x \ is \ a \ two \ digit \ number, \ the \ sum \ of \ the \ digits \ is \ 8 \}

Answer: H = \{ 17, 26, 35, 44, 53, 62, 71, 80 \}

Note: If the first digit is x , then the second digit is (8-x) . So now try the values of x as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \cdots x cannot be 0 as it is a two digit number. Also by same logic, x cannot be greater than 8 either. Hence x can only be 1, 2, 3, 4, 5, 6, 7, and 8 only.

(vii) I = \{ x | x \in N, x \ is \ divisible \ by \ both \ 4 \ and \ 6 \ and \ x \leq 60 \}

Answer: I = \{ 12, 24, 36, 48, 60 \}

Note: x \leq 60 means that x = \{ 1, 2, 3 \cdots 60 \} . Of these numbers x needs to be divisible both by 4 and 6 .

(viii) J = \{ x | x =  (1/n) , n \in N \ and \ n \leq 5 \}

Answer: J = \{1 , \cdots \}

Note: n \in N and n \leq 5 which means that n is 1, 2, 3, 4, 5 . Now substitute the values.

(ix)  L = \{ x | x \ is \ a \ letter \ of \ the \ word \ 'careless' \}

Answer: L = \{ c, a, r, e, l, s \}

Note: In careless, the alphabet e and s are repeated. We only need to take into account distinct elements only.

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Question 4: Describe the following sets in set builder form

(i) A = \{ 5, 6, 7, 8, 9, 10, 11, 12\}

Answer:  A = \{x | x \in N, 4 < x < 13 \} or A = \{x | x \in N, 5 \leq x \leq 12 \}

(ii) B = \{1, 2, 3, 4, 6, 8, 12, 16, 24, 48 \}

Answer:  B = \{x | x \ is \ a \ factor \ of \ 48 \}

(iii)  C = \{11, 13, 17, 19, 23, 29, 31, 37 \}

Answer:  C = \{x | x \ is \ a \ prime \ number, \ x \in \ N \ and \ 10 < x < 30 \}

(iv) D = \{21, 23, 25, 27, 29, 31, 33, 35, 37 \}

Answer:  D = \{x | x \ is \ an \ odd \ number, \ 20 < x < 38 \} or

(v) D = \{x | x = (2n+19), n \in N, \ and \ 1 \leq n \leq 9 \}

(vi) I = \{9, 16, 25, 36, 49, 64, 81, 100 \}

Answer: I = \{x | x=n2, n \in N, \ and\ 3 \leq n \leq 10 \}

Note: If you notice, the numbers are square of 3, 4, 5, 6, 7, 8, 9, 10 .

(vii) J = \{ -2, 2 \}

Answer:  J = \{x | x \in Z, x^2 = 4 \}

Note: Integers (Z): Positive and negative counting numbers, as well as zero: \{ ... , -2, -1, 0, 1, 2 , ... \}

(viii) x^2=4 which is x = \sqrt{4} which means x can be 2 or -2

(ix) K = \{0 \}

Answer:  K = \{x | x = 0 \}

(x)  L = \{ \}

Answer: L = \{x | x \in N, x \neq x \}

(xi)  M = \{a, b, c, d, e, f, g, h, i \}

Answer: M = \{x | x \ is \ an \ English \ alphabet \ which \ precedes \ J \}

(xii)  P = \{2/7, 3/8,  4/9,  5/10,  6/11,  7/12,  8/13,  9/14 \}

Answer:  P = \{x | x = , n \in N, 2 \leq n \leq 9 \}

(xiii) S = \{Atlantic, Artic \}

Answer:  S = \{x | x \ is \ an \ ocean \ name \ that \ starts \ with \ A \}

(xiv) T = \{Mars, Mercury \}

Answer:  T = \{x | \ x \ is \ a \ planet \ whose \ name \ starts \ with \ an \ M \}

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Question 5: Separate finite and infinite sets from the following:

(i)  Set of leaves on a tree

Answer: Finite. This is because in this case, the process of counting the leaves would surely come to an end.

(ii) Set of all counting numbers

Answer: Infinite. This is because there is no end to the numbers.

(iii) \{x | x \in N, x > 1000 \}

Answer: Infinite. This is because there is no end to the numbers since x > 1000 .

(iv)  \{x | x \in W, x < 5000 \}

Answer: Finite. W is whole numbers which are all natural numbers including 0. Since x is less than 5000, there are finite numbers to be counted.

(v) \{x | x \in Z, x < 4 \}

Answer: Infinite. Z is an integer. Integers can be negative numbers too. So in this case, though x is limited to less than 4 on the positive scale, it can go to infinity on the negative scale.

(vi) Set of all triangles in a plane

Answer: Infinite. This is because there can be uncountable number of triangles in a plane.

(vii) Set of all points on a circumference of a circle

Answer: Infinite. This is because there can be uncountable number of points on a circumference of a circle.

(viii) \{ 1, 2, 3, 1, 2, 3, 1, 2, 3, ... \}

Answer: Infinite. This is because the set is not ending. It will keep having 1, 2, 3 repeated again and again

(ix) \{x | x \in Q, 2 < x< 3 \}

Answer: Infinite.

Note: Rational numbers (Q): Numbers that can be expressed as a ratio of an integer to a non-zero integer. All integers are rational, but the converse is not true.

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Question 6: Which of the following are empty sets?

(i) A = \{ x | x \in N, x + 5 = 5 \}

Answer: Empty Set

Note: Natural Number N = \{ 1, 2, 3, 4, 5, ..., \} . Since x+5 = 5 is given, it means, x = 0 . But 0 does not belong to N and hence the set it empty or null.

(ii) B = \{ x | x \in N, 2x + 3 = 6 \}

Answer: Empty Set

Note: Natural Number N = \{ 1, 2, 3, 4, 5, ..., \} . Since 2x+3 = 6 is given, it means, x = 1.5 . But 1.5 does not belong to N and hence the set it empty or null.

(iii) C = \{ x | x \in W, x + 2 < 5 \}

Answer: Not an Empty Set

Note: Whole Number W = \{ 0,1, 2, 3, 4, 5, ..., \} . Since x+2 < 5 is given, it means, x < 3 . Therefore x can be 0, 1, 2 and hence it is not an Empty Set.

(iv)  D = \{ x | x \in N, 1 < x \leq 2 \}

Answer: Not an Empty Set

Note: Natural Number N = \{ 1, 2, 3, 4, 5, ..., \} . Since 1 < x \leq2 , it means that x can be 2 . Hence D is not an Empty Set.

(v)  E = \{ x | x \in N, x^2+ 4 = 0 \}

Answer: Empty Set

Note: Natural Number N = \{ 1, 2, 3, 4, 5, ..., \} x^2 + 4 = 0 means that x^2 is - 4 which is not possible.

(vi)  F = \{ x | x \ is \ a \ prime \ number, 90 < x< 96 \}

Answer: Empty Set

Note: If 90 < x< 96 , this means x can be 91, 92, 93, 94, 95 . All these numbers are divisible by a number. E.g. 91 is divisible by 7, 92 by 2, 93 by 3 and so on.

(vii) G = \{x | x \ is \ an \ even \ prime \}

Answer: Not an Empty Set.

Note: 2 is an even prime number.

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Question 7: Which of the following are pairs of equivalent sets?

Note: Two finite sets A and B are said to be equivalent, if n(A) = n(B), that is they have the same number of elements. An equivalent set is simply a set with an equal number of elements. The sets do not have to have the same exact elements, just the same number of elements.

(i) A = \{2, 3, 5, 7 \} and B = \{x : x \ is \ a \ whole \ number, \ x < 3 \}

Answer: No

Note: A = \{2, 3, 5, 7 \} and B = \{0, 1, 2 \}   Therefore  n(A) = 4 while n(B) = 3

(ii) C = \{x : x + 2 = 2 \} and D = \phi

Answer: No

Note: C = \{0 \} and D = \{ \} Therefore n(C) = 1 while n(D) =  null

(iii) E = \{x : x \ is \ a \ natural \ number, \ x < 4 \} and F = \{x : x \ is \ a \ whole \ number, x <  3 \}

Answer: Yes

Note: E = \{ 1, 2, 3 \} and F = \{ 0, 1, 2 \} Therefore n(E) = 3 while n(F) = 3 Hence E and F are equivalent sets.

(iv) G = \{x : x \ is \ an \ integer, - 3 < x < 3 \} and H = \{x : x is a factor of 16 \}

Answer: Yes

Note: G = \{-2, -1, 0, 1, 2 \} and H = \{1, 2, 4, 8, 16 \} Therefore n(G) = 5 and n(H) = 5 Hence they are equivalent sets

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Question 8: State whether the following statements are true or false.

(i)  \{ a, b, c, 1, 2, 3 \} is not a set

Answer: False. It is a set. It contains elements.

(ii)  \{ 5, 7, 9 \} = \{ 9, 5, 7 \}

Answer: True. Both sets contain the same elements just in different order. The order of elements does not matter.

(iii) \{ x | x \in W, x + 8 = 8 \} is a singleton set

Answer: True. The only element in the set would be 0 . Hence it is a singleton set.

Note: a singleton, also known as a unit set, is a set with exactly one element. For example, the set {0} is a singleton. The term is also used for a 1-tuple (a sequence with one element).

(iv)  \{ x | x \in W, x < 0 \} = f

Answer: True. The set is a null set as there is no element.

(v) \{ x | x \in N, x + 5 = 3 \} = f

Answer: True. x = -2 but x is a natural number. Hence a null set.

(vi) \{ x | x \in N, 3 < x \leq 4 \} = f

Answer: False.  There is a valid element in the set which is 4 .

(vii) If A = \{ x | x \ is \ a \ letter \ of \ the \ work, \ 'Meerut' \} then n(A)=6

Answer: False. n(A)=5

(viii) If A = \{ x | x \in N, 8 < x < 13 \} and B = \{ x | x \in Z, -3 < = x <1 \} , then n(A) = n(B)

Answer: True. A = \{ 9, 10, 11, 12 \} . Therefore n(A) = 4 . B = \{-3, -2, -1, 0 \} Therefore n(B) = 4 . Hence n(A) = n(B)

(ix)  If n(A) = n(B) , then A = B

Answer: False. Just because the number of elements in the sets A and B are equal, does not mean that the sets are also equivalent.

(x) If B is a set of all consonants in English Alphabets, then n(B)=21

Answer: True

Note: The word consonant is also used to refer to a letter of an alphabet that denotes a consonant sound. The 21 consonant letters in the English alphabet are B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, X, Z , and usually W and Y .

(xi) \{ x | x \ is \ a \ prime \ factor \ of \ 24 \} = \{ 2, 3, 4, 6, 8, 12, 24 \}

Answer: False. Prime factors of 24 are 1, 2 , and 3