**1** **Which of the following are sets? Justify our answer.$\\$ (i) The collection of all months of a year beginning with the letter J.$\\$ (ii) The collection of ten most talented writers of India.$\\$ (iii) A team of eleven best-cricket batsmen of the world.$\\$ (iv) The collection of all boys in your class.$\\$ (v) The collection of all natural numbers less than $100$.$\\$ (vi) A collection of novels written by the writer Munshi Prem Chand.$\\$ (vii) The collection of all even integers.$\\$ (viii) The collection of questions in this chapter.$\\$ (ix) A collection of most dangerous animals of the world.**

(i) The collection of all months of a year beginning with the letter J is a well-defined collection of objects because one can definitely identity a month that belongs to this collection. Hence, this collection is a set.$\\$ (ii) The collection of ten most talented writer of India is not a well-defined collection because the criteria for determining a writer’s talent vary from person to person. Hence, this collection is not a set.$\\$ (iii) A team of eleven best cricket batsmen of the world is not a well-defined collection because the criteria for determining a batsman’s talent may vary from person to person. Hence, this collection is not a set.$\\$ (iv) The collection of all boys in your class is a well-defined collection because you can definitely identify a boy who belongs to this collection. Hence, this collection is a set.$\\$ (v) The collection of all natural numbers less than 100 is a well-defined collection because one can definitely identify a number that belongs to this collection. Hence, this collection is a set.$\\$ (vi) A collection of novels written by the writer Munshi Prem Chand is a well-defined collection because one can definitely identify a book that belongs to this collection. Hence, this collection is a set.$\\$ (vii) The collection of all even integers is a well-defined collection because one can definitely identify an even integer that belongs to this collection. Hence, this collection is a set.$\\$ (viii) The collection of questions in this chapter is a well-defined collection because one can definitely identify a question that belongs to this chapter. Hence, this collection is a set.$\\$ (ix) The collection of most dangerous animals of the world is not a well-defined collection because the criteria for determining the dangerousness of an animal can vary from person to person. Hence, this collection is not a set.$\\$

**2** **Let $A =\{ 1, 2,3, 4,5,6 \}$ . Insert the appropriate symbol $ \in $or$\notin $ in the blank spaces: $(i) 5...A\\ (ii) 8...A\\ (iii) 0...A\\ (iv) 4...A\\ (v) 2...A\\ (vi) 10...A$**

$(i) 5 \in A\\ (ii) 8 \notin A\\ (iii) 0 \notin A\\ (iv) 4 \in A\\ (v) 2 \in A\\ (vi) 10 \notin A$

**3** **Write the following sets in roster form:$\\$ (i) $A =\{ x : x$ is an integer and $- 3 < x < 7 \} .$$\\$ (ii) $B =\{ x : x$ is a natural number less than $6 \}.$$\\$ (iii) $C =\{ x : x$ is a two-digit natural number such that sum of its digitsis $8 \}$ .$\\$ (iv)$ D =\{ x : x$ is a prime number which is divisor of $60 \}.$$\\$ (v) $E$ = The set of all letters in the world TRIGONOMETRY$\\$ (vi)$ F$ = The set of all letters in the word BETTER.**

(i) $A =\{ x : x$ is an integer and $- 3 < x < 7 \} .$$\\$ The elements of this set are $-2,-1,0,1,2,3,4,5$ and $6 $ only Therefore, the given set can be written in roster form as $A=\{-2,-1,0,1,2,3,4,5,6\}$$\\$ (ii) $B =\{ x : x$ is a natural number less than $6 \}.$$\\$ The elements of this set are $1, 2, 3, 4$ and $5$ only. Therefore, the given set can be written in roster form as $B = \{1, 2, 3, 4, 5\}$$\\$ (iii) $C =\{ x : x$ is a two-digit natural number such that sum of its digitsis $8 \}$ .$\\$ The elements of this set are $17, 26, 35, 44, 53, 62, 71$ and $80$ only. Therefore, this set can be written in roster form as $C=\{17, 26, 35, 44, 53, 62, 71,80\}$$\\$ (iv)$ D =\{ x : x$ is a prime number which is divisor of $60 \}.$$\\$ $ \begin{array}{|c|c|c|c|} \hline 2 & 60 \hline 2 & 30 \hline 3 & 15 \hline & 5 \hline \end{array}$ $\\$ $\therefore 60=2*2*3*5$$\\$ The elements of this set are $2, 3 $ and $5$ only. Therefore, this set can be written in roster form as $ D=\{2,3,5 \}$

(v) E = The set of all letters in the word TRIGONOMETRY$\\$ There are $12$ letters in the word TRIGONOMETRY, out of which letters T, R and O are repeated Therefore, this set can be written in roster form as $E = \{T, R, I, G, O, N, M, E, Y\}$$\\$ (vi) F = The set of all letters in the word BETTER$\\$ There are 6 letters in the word BETTER, out of which letters E and T are repeated. Therefore, this set can be written in roster form as $F =\{ B , E , T , R \}.$

**4** **Write the following sets in the set-builder form:$\\$ (i) $(3, 6, 9, 12)$$\\$ (ii) $\{2, 4, 8, 16, 32\}$$\\$ (iii) $\{5, 25, 125, 625\}$$\\$ (iv) $\{2, 4, 6 ...\}$$\\$ (v) $\{1, 4, 9 ... 100\}$**

(i) A=$\{$ x : x is an odd natural number$\}=\{1,3,5,7,9,...\}$$\\$ (ii) B =$\{$ x : x is an integer;$-\dfrac{1}{2} < x < \dfrac{9}{2}\}$$\\$ It can be seen that $-\dfrac{1}{2}=-0.5 $ and $ \dfrac{9}{2}=4.5 $$\\$ $\therefore B=\{0,1,2,3,4\}$$\\$ (iii) C =$\{$x : x isan integer; x$^2 \leq 4$$\\$ It can be seen that $\\$ $(-1)^2=1 \leq 4;\\ (-2)^2 =4 \leq 4;\\ (-3)^2=9 > 4 \\ 0^2=0 \leq 4\\ 1^2=1 \leq 4\\ 2^2=4 \leq 4\\ 3^2=9> 4\\ \therefore C=\{-2,-1,0,1,2\}$$\\$ (iv) D =$\{$ x : x isa letter in the word"LOYAL" $\}=\{L,O,Y,A\}$$\\$ (v) E =$\{$ x : x isa month of a year not having $31$ days $\\$ $=\{\text{February, April, June,Septermber, November}\}$$\\$ (vi) F=$\{$ x : x isa consonant in the English alphabet which precedes k $\}$$\\$ $=\{b,c,d,f,g,h,j\}$

**5** **List all the elements of the following sets:$\\$ (i) A=$\{$ x : x is an odd natural number$\}$$\\$ (ii) B =$\{$ x : x is an integer;$-\dfrac{1}{2} < x < \dfrac{9}{2}\}$$\\$ (iii) C =$\{$x : x isan integer; x$^2 \leq 4$$\\$ (iv) D =$\{$ x : x isa letter in the word"LOYAL" $\}$$\\$ (v) E =$\{$ x : x isa month of a year not having $31$ days $\\$ (vi) F=$\{$ x : x isa consonant in the English alphabet which precedes k $\}$**

(i) A=$\{$ x : x is an odd natural number$\}=\{1,3,5,7,9,...\}$$\\$ (ii) B =$\{$ x : x is an integer;$-\dfrac{1}{2} < x < \dfrac{9}{2}\}$$\\$ It can be seen that $-\dfrac{1}{2}=-0.5 $ and $ \dfrac{9}{2}=4.5 $$\\$ $\therefore B=\{0,1,2,3,4\}$$\\$ (iii) C =$\{$x : x isan integer; x$^2 \leq 4$$\\$ It can be seen that $\\$ $(-1)^2=1 \leq 4;\\ (-2)^2 =4 \leq 4;\\ (-3)^2=9 > 4 \\ 0^2=0 \leq 4\\ 1^2=1 \leq 4\\ 2^2=4 \leq 4\\ 3^2=9> 4\\ \therefore C=\{-2,-1,0,1,2\}$$\\$ (iv) D =$\{$ x : x isa letter in the word"LOYAL" $\}=\{L,O,Y,A\}$$\\$ (v) E =$\{$ x : x isa month of a year not having $31$ days $\\$ $=\{\text{February, April, June,Septermber, November}\}$$\\$ (vi) F=$\{$ x : x isa consonant in the English alphabet which precedes k $\}$$\\$ $=\{b,c,d,f,g,h,j\}$

**6** **If$(\dfrac{x}{3}+1,y-\dfrac{2}{3})=(\dfrac{5}{3},\dfrac{1}{3}),$ find the values of x and y.**

It is given that set A has 3 elements and the elements of set $B$ are $3, 4,$ and $5.$$\\$ $\implies $ Number of elements in set $B = 3$$\\$ Number of elements in $( A * B )$$\\$ = (Number of elements in A) * (Number of elements in B) $= 3 * 3 = 9$$\\$ Thus, the number of elements in (A * B } in $9.$

**7** **Match each of the set on the left in the roster form with the same set on the right described in set-builder form:$\\$ (i) $\{1, 2, 3, 6\} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (a) $\{$ x : x is a primenumber anda divisor of $6\}$$\\$ (ii) $\{2, 3\} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (b) $\{$ x : x is an odd natural number less than $10\}$$\\$ (iii)$ \{M, A, T, H, E, I, C, S\} \ \ $ (c) $\{$ x : x is natural number and divisor of $6 \}$$\\$ (iv) $\{1, 3, 5, 7, 9\} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (d) $\{$ x : x is a letter of the word MATHEMATICS $\}$**

(i) All the elements of this set are natural numbers as well as the divisors of $6.$ Therefore, (i) matches with (c).$\\$ (ii) It can be seen that $2$ and $3$ are prime numbers. They are also the divisors of $6.$ Therefore, (ii) matches with (a).$\\$ (iii) All the elements of this set are letters of the word MATHEMATICS. Therefore, (iii) matches with (d).$\\$ (iv) All the elements of this set are odd natural numbers less than $10.$ Therefore, (iv) matches with (b).

**8** **Which of the following are examples of the null set $\\$ (i) Set of odd natural numbers divisible by $2$$\\$ (ii) Set of even prime numbers$\\$ (iii) $\{$x : x is a natural numbers, x $ < 5$ and x $> 7\}$$\\$ (iv) $\{$y : y is a point common to any two parallel lines $\}$**

(i) A set of odd natural numbers divisible by $2$ is a null set because no odd number is divisible by $2.$$\\$ (ii) A set of even prime numbers is not a null set because $2$ is an even prime number.$\\$ (iii) $\{$x : x is a natural number, x $< 5$ and x $> 7 \}$ is a null set because a number cannot be simultaneously less than $5$ and greater than $7.$$\\$ (vi) $\{$ y : y is a point common to any two parallel lines $\}$ is a null set because parallel lines do not intersect. Hence, they have no common point.

**9** **Which of the following sets are finite or infinite$\\$ (i) The set of months of a year$\\$ (ii) $\{1, 2, 3....\}$$\\$ (iii) $\{1, 2, 3 ... 99, 100\}$ (iv) The set of positive integers greater than $100$$\\$ (v) The set of prime numbers less than $99$**

(i) The set of months of a year is a finite set because it has $12$ elements.$\\$ (ii) $\{1, 2, 3 ...\}$ is an infinite set as it has infinite number of natural numbers.$\\$ (iii) $\{1, 2, 3... 99, 100\}$ is a finite set because the numbers from $1$ to $100$ are finite in number.$\\$ (iv) The set of positive integers greater than $100$ is an infinite set because positive integers greater than $100$ are infinite in number.$\\$ (v) The set of prime numbers less than $99$ is a finite set because prime numbers less than $99$ are finite in number.

**10** **State whether each of the following set is finite or infinite:$\\$ (i) The set of lines which are parallel to the x-axis$\\$ (ii) The set of letters in the English alphabet$\\$ (iii) The set of numbers which are multiple of $5$$\\$ (iv) The set of animals living on the earth$\\$ (v) The set of circles passing through the origin $(0, 0)$**

(i) The set of lines which are parallel to the x-axis is an infinite set because lines parallel to the x-axis are infinite in number.$\\$ (ii) The set of letters in the English alphabet is a finite set because it has $26$ elements.$\\$ (iii) The set of numbers which are multiple of $5$ is an infinite set because multiple of $5$ are infinite in number.$\\$ (iv) The set of animals living on the earth is a finite set because the number of animals living on the earth is finite (although it is quite a big number).$\\$ (v) The set of circles passing through the origin $(0, 0)$ is an finite set because infinite number of circles can pass through the origin.

**11** **In the following, state whether A = B or not:$\\$ (i) A =$\{ a , b , c , d \}$ ; B =$\{ d , c , b , a \}$$\\$ (ii) A=$\{ 4,8,12,16 \}$ ; B =$\{ 8, 4,16,18 \}$$\\$ (iii) A=$\{ 2, 4,6,8,10 \}$ ; B =$\{$ x : x is positive even integer and x $\leq 10 \}$$\\$ (iv) A=$\{$ x : x is a multiple of $10 \}$ ; B=$\{ 10,15, 20, 25,30....... \}$**

(i) A=$\{ a , b , c , d \}$ ; B =$\{ d , c , b , a \}$$\\$ The order in which the elements of a set are listed is not significant. $\therefore A = B$$\\$ (ii) A $=\{4,8,12,16 \}$ ; B=$\{ 8, 4,16,18 \}$$\\$ It can be seen that $12 \in$ A but $12 \notin$ B .$\\$ $\therefore A \neq B$$\\$ (iii) A $=\{ 2, 4,6,8,10 \}$$\\$ B =$\{$ x : x is a positive even integer and x $\leq 10 \}$$\\$ = $\{2, 4, 6, 8, 10\}$$\\$ $\therefore A = B$$\\$ (iv) A $=\{$ x : x isa multiple of $10\}$$\\$ B =$\{10,15, 20, 25,30.......\}$$\\$ It can be seen that $15 \in B$ but $15 \notin A .$$\\$ $\therefore A \neq B$

**12** **Are the following pair of sets equal? Give reasons.$\\$ (i) A $=\{ 2,3 \}$ ; B =$\{$ x : x is solution of $x^ 2+ 5 x + 6 = 0\}$$\\$ (ii) A$=\{$ x : x is a letter in the word FOLLOW $\}$ ; B $=\{$ y : y is a letter in the word WOLF $\}$**

(i) A $=\{ 2,3 \}$ ; B =$\{$ x : x is a solution of $x^ 2 + 5 x + 6 = 0\}$$\\$ The equation $x ^2 + 5 x + 6 = 0$ can be solved as:$\\$ $x(x+3)+2(x+3)=0\\ (x+2)(x+3)=0\\ x=-2 \ \ 0r \ \ \ x=-3\\ \therefore A=\{2,3\};B=\{-2,-3\} \\ \therefore A\neq B$$\\$ (ii)A=$\{$x : x is a letter in the word FOLLOW $\}=\{$ F,O, L, W $\}$$\\$ B$=\{$ y : y is a letter in the word WOLF$\}=\{$ W,O, L, F $\}$$\\$ The order in which the elements of a set are listed is not significant. $\therefore A = B$

**13** **From the sets given below, select equal sets:$\\$ $A =\{2, 4,8,12 \} , B =\{ 1, 2,3, 4 \} ,\\ C =\{ 4,8,12,14 \} , D =\{ 3,1, 4, 2 \}\\ E =\{ -1,1 \} , F =\{ 0, a \} ,\\ G =\{ 1, -1 \} , H =\{ 0,1 \}$**

$A =\{2, 4,8,12 \} , B =\{ 1, 2,3, 4 \} ,\\ C =\{ 4,8,12,14 \} , D =\{ 3,1, 4, 2 \}\\ E =\{ -1,1 \} , F =\{ 0, a \} ,\\ G =\{ 1, -1 \} , H =\{ 0,1 \}$$\\$ It can be seen that $\\$ $8\in A,8\notin B,8\notin D,8\notin E,8\notin F,8 \notin G,8\notin H\\ \implies A\neq B,A \neq D,A\neq E,A\neq F,A \neq G ,A \neq H$$\\$ Also, $ 2\in A,2 \notin C\\ \therefore A\neq C\\ 3 \in B,3 \notin D,3\notin E,3\notin F,3 \notin G ,3 \notin H \\ \therefore B\neq C,B\neq E,B\neq F,B\neq G,B\neq H\\ 12 \in C,12\notin D,12 \notin E ,12 \notin F ,12 \notin G ,12 \notin H\\ \therefore C\neq D,C\neq E,C\neq F,C\neq G,C\neq H\\ 4 \in D ,4 \notin E ,4 \notin F ,4 \notin G ,4 \notin H\\ \therefore D \neq E , D \neq F , D \neq G , D \neq H$$\\$ Similarly,$ E \neq F , E \neq G , E \neq H\\ F \neq G , F \neq H , G \neq H $$\\$ The order in which the elements of a set are listed is not significant. $\therefore B = D and E = G$$\\$ Hence, among the given sets, B = D and E = G.

**14** **Make correct statements by filling in the symbols $\subset$ or $\not\subset $ in the blank spaces:$\\$ (i) $\{ 2,3, 4 \} ... \{ 1, 2,3, 4,5 \}$$\\$ (ii) $\{ a , b , c \} ... \{ b , c , d \}$$\\$ (iii) $\{$ x : x is a student of class $XI$ of your school $\} ... \{$x : x student of your school $\}$$\\$ (iv) $\{$ x : x is a circle in the plane $\}... \{$ x : x is a circle in the same plane with radius $1$ unit $\}$$\\$ (v) $\{$ x : x is a triangle in a plane $\}... \{$ x : x is a rectangle in the plane $\}$$\\$ (vi) $\{$ x : x is an equilateral triangle in a plane $\}... \{$ x : x is a triangle in the same plane $\}$$\\$ (vii) $\{$ x : x is an even natural number $\} ... \{$ x : x is an integer \}$**

(i) $\{ 2,3, 4 \} \subset \{ 1, 2,3, 4,5 \}$$\\$ (ii) $\{ a , b , c \} \not \subset \{ b , c , d \}$$\\$ (iii) $\{$ x : x is a student of class $XI$ of your school $\} \subset \{$x : x student of your school $\}$$\\$ (iv) $\{$ x : x is a circle in the plane $\} \not \subset \{$ x : x is a circle in the same plane with radius $1$ unit $\}$$\\$ (v) $\{$ x : x is a triangle in a plane $\} \not \subset \{$ x : x is a rectangle in the plane $\}$$\\$ (vi) $\{$ x : x is an equilateral triangle in a plane $\} \subset \{$ x : x is a triangle in the same plane $\}$$\\$ (vii) $\{$ x : x is an even natural number $\} \subset \{$ x : x is an integer \}$

**15** **Examine whether the following statements are true or false:$\\$ (i) $\{ a , b \} \not \subset \{ b , c , a \}$$\\$ (ii) $\{ a , e \} \subset \{$ x : x is a vowel in the English alphabet $\}$$\\$ (iii) $\{ 1, 2,3\} \subset \{ 1,3,5\}$$\\$ (iv) $\{ a \} \subset \{a , b , c \}$$\\$ (v) $\{ a \}\in \{ a , b , c \}$$\\$ (vi) $\{$ x : x is an even natural number less than $6 \} \subset \{$ x : x is a natural number which divide $36 \}$**

(i) False. Each element of $\{ a , b \}$ is also an element of $\{ b , c , a \}$ .$\\$ (ii) True, a , e are two vowels of the English alphabet.$\\$ (iii) False. $2 \in 1, 2,3 \}$ ; however, $2 \notin 1,3,5 \}$$\\$ (iv) True. Each element of $\{a\}$ is also an element of $\{a, b, c\}$.$\\$ (v) False. The element of $\{a, b, c\}$ are $a, b, c.$ Therefore, $\{a \} \subset \{a , b , c \}$ (vi) True. $\{$ x : x is an even natural number less than $6 \}=\{ 2, 4 \}$$\\$ $\{$x : x is a natural number which divides $36\}=\{ 1, 2,3, 4,6,9,12,18,36 \}$

**16** **Let $A =\{ 1, 2, \{ 3, 4 \} ,5 \} .$ Which of the following statements are incorrect and why?$\\$ $(i) \{ 3, 4 \} \subset A\\ (ii) \{3,4\}\in A\\ (iii)\{\{3,4\}\}\subset A\\ (iv)1\notin A\\ (v)1 \subset A\\ (vi) \{1,2,5\} \subset A\\ (vii) \{1,2,5\}\in A\\ (viii) \{1,2,3\} \subset A\\ (ix) \varnothing \notin A\\ (x) \varnothing \subset A\\ (xi) \{\varnothing \} \subset A$**

A $=\{ 1, 2,\{3, 4 \} ,5\}$$\\$ (i) The statement $\{ 3, 4 \} \subset $ A is incorrect because $3 \in \{ 3, 4 \}$ ; however, $3 \notin $ A .$\\$ (ii) The statement $\{ 3, 4\} \notin $ A is correct because $\{ 3, 4 \}$ is an element of A.$\\$ (iii) The statement $\{ \{ 3, 4 \} \} \subset $ A is correct because $\{ 3, 4\}\notin \{ \{ 3, 4 \} \}$ and $\{ 3, 4 \}\in $ A .$\\$ (iv) The statement $1 \notin $ A is correct because $1$ is an element of A.$\\$ (v) The statement $1 \subset $ A is incorrect because an element of a set can never be a subset of itself.$\\$ (vi) The statement $\{ 1, 2,5\} \subset $ A is correct because each element of $\{ 1, 2,5 \}$ is also an element of A.$\\$ (vii) The statement $\{ 1, 2,5 \} \notin $ A is incorrect because $\{1, 2,5 \}$ is not an element of A. (viii) The statement $\{ 1, 2,5\} \subset $ A is incorrect because $3 \notin \{ 1, 2,3 \}$ ; however, $3 \notin A .$$\\$ (ix) The statement $\varnothing \notin $ A is incorrect because $\varnothing $ is not an element of A.$\\$ (x) The statement $\varnothing \subset $ A is correct because $\varnothing $ is a subset of every set.$\\$ (xi) The statement $\{ \varnothing \} \subset $ A is incorrect because, $\varnothing $ is a subset of A and it is not an element of A.

**17** **Write down all the subsets of the following sets:$\\$ (i)$\{ a \}$$\\$ (ii)$\{a,b\}$$\\$ (iii)$\{1,2,3\}$$\\$ (iv)$\varnothing $**

(i) The subsets of $\{a \}$ are $ \varnothing $ and $\{ a \} .$$\\$ (ii) The subsets $\{a , b \}$ are $\varnothing , \{ a \} , \{ b \} ,$ and $\{ a , b \}$ .$\\$ (iii) The subsets of $\{ 1, 2,3 \}$ are $\varnothing , \{ 1 \} , \{ 2 \} , \{ 3 \} , \{ 1, 2 \} , \{ 2,3 \} , \{ 1,3 \}$ and $\{ 1, 2,3 \}$ .$\\$ (iv) The only subset of $\varnothing $ is $\varnothing $ .

**18** **How many elements has $P ( A )$ , if $A = \varnothing ? $**

We know that if A is a set with m elements i.e., $n(A)=m,$ , then $n[p(A)]=2^m$$\\$ If A $=\varnothing $ , then $n ( A )= 0 .$$\\$ $\therefore n [ P ( A)]= 2 ^0 = 1$$\\$ Hence, $P ( A ) $ has one element.

**19** **Write the following as intervals:$\\$ (i) $\{ x : x \notin R , - 4 < x \leq 6 \}$$\\$ (ii) $\{ x : x \notin R , - 12 < x <- 10 \}$$\\$ (iii) $\{ x : x \notin R ,0 \leq x < 7 \}$$\\$ (iv) $\{ x : x \notin R ,3 \leq x \leq 4 \}$**

(i) $\{ x : x \notin R , - 4 < x \leq 6 \}=(-4,6)$$\\$ (ii) $\{ x : x \notin R , - 12 < x <- 10 \}=(-12,-10)$$\\$ (iii) $\{ x : x \notin R ,0 \leq x < 7 \}=(0,7)$$\\$ (iv) $\{ x : x \notin R ,3 \leq x \leq 4 \}=(3,4)$

**20** **Write the following intervals in set-builder form: $\\$ (i)$(-3,0)$$\\$ (ii)$[6,12)$$\\$ (iii)$(6,12)$$\\$ (iv)$(-23,5)$$\\$**

(i)$(-3,0)=\{x:x\notin R,-3 < x < 0 \}$$\\$ (ii)$[6,12)=\{x:x \notin R,6 \leq x \leq 12 \}$$\\$ (iii)$(6,12)=\{x:x\notin R,6 < x \leq 12\}$$\\$ (iv)$(-23,5)=\{ x: x \notin R,-23 \leq x < 5 \}$$\\$

**21** **What universal set (s) would you propose for each of the following:$\\$ (i) The set of right triangles$\\$ (ii) The set of isosceles triangles**

(i) For the set of right triangles, the universal set can be the set of triangles or the set of polygons.$\\$ (ii) For the set of isosceles triangles, the universal set can be the set of triangles or the set of polygons or the set of two-dimensional figures.

**22** **Given the sets $A=\{1,3,5\},B=\{2,4,6\}$ and $ C=\{0,2,4,6,8\},$which of the following may be considered as universals set (s) for all the three sets $A, B$ and $C.$$\\$ (i) $\{ 0,1, 2,3, 4,5,6 \}$$\\$ (ii) $\varnothing $$\\$ (iii) $\{ 0,1, 2,3, 4,5,6,7,8,9,10 \}$$\\$ (iv) $\{ 1, 2,3, 4,5,6,7,8 \}$**