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.

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

(ii)  All the rivers flowing in UP

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

$A = \{ \text{ Ganga, Yamuna} \cdots \}$

(iii)  All the planets in our solar system

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

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

(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

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

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

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

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

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

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

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

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

Answer:  $D = \{x | x \text{ is an odd number }, \ 20 < x < 38 \}$ 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: $M = \{x | x \text{ is an English alphabet which precedes J} \}$

(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) $S = \{ \text{ Atlantic, Artic} \}$

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

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

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

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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) $\{ 1, 2, 3, 1, 2, 3, 1, 2, 3, \cdots \}$

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

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

Note: Rational numbers (Q): Numbers that can be expressed as a 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) $A = \{ x | x \in N, x + 5 = 5 \}$

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

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

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

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

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

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

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) $G = \{x | x \text{ is an even prime} \}$

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 $B = \{x : x \text{ is a whole number,} \ x < 3 \}$

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 $D = \phi$

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

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

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

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 $A = \{ x | x \text{ is a letter of the work, 'Meerut'} \}$ 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 $n(B)=21$

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) $\{ x | x \text{ is a prime factor of} 24 \} = \{ 2, 3, 4, 6, 8, 12, 24 \}$
Answer: False. Prime factors of $24$ are $1, 2$, and $3$