# Sets

## Class 11 NCERT

### NCERT

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.

##### Solution :

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

##### Solution :

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

(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 ...\}$$\\ (v) \{1, 4, 9 ... 100\} ##### Solution : (i) A=\{ x : x is an odd natural number\}=\{1,3,5,7,9,...\}$$\\$ (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 $\}$

##### Solution :

(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 $\}$

##### Solution :

(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 $99$

##### Solution :

(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.

##### Solution :

(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 $\}$

##### Solution :

(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:$\\$ $A =\{2, 4,8,12 \} , B =\{ 1, 2,3, 4 \} ,\\ C =\{ 4,8,12,14 \} , D =\{ 3,1, 4, 2 \}\\ E =\{ -1,1 \} , F =\{ 0, a \} ,\\ G =\{ 1, -1 \} , H =\{ 0,1 \}$

##### Solution :

(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?$\\$ $(i) \{ 3, 4 \} \subset A\\ (ii) \{3,4\}\in A\\ (iii)\{\{3,4\}\}\subset A\\ (iv)1\notin A\\ (v)1 \subset A\\ (vi) \{1,2,5\} \subset A\\ (vii) \{1,2,5\}\in A\\ (viii) \{1,2,3\} \subset A\\ (ix) \varnothing \notin A\\ (x) \varnothing \subset A\\ (xi) \{\varnothing \} \subset A$

##### Solution :

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\}$$\\ (iv)\varnothing ##### Solution : (i) The subsets of \{a \} are \varnothing and \{ a \} .$$\\$ (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 $A = \varnothing ?$

##### Solution :

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) $\{ x : x \notin R ,3 \leq x \leq 4 \}$