Sets

(i) The collection of all the months of a year beginning with the letter J.

Sol. The months of a year beginning with the letter J are January, June and July. The collection is well defined.

∴ It is a set.

(ii) The collection of ten most talented writers of India.

Sol. Whether a writer is among the ten most talented writers of India depends on a person's opinion, so the collection is not well defined.

∴ It is not a set.

(iii) A team of eleven best-cricket batsmen of the world.

Sol. Whether a batsman is among the eleven best batsmen of the world depends on a person's opinion, so the collection is not well defined.

∴ It is not a set.

(iv) The collection of all boys in your class.

Sol. For any student we can decide definitely whether the student is a boy of your class or not. The collection is well defined.

∴ It is a set.

(v) The collection of all natural numbers less than 100.

Sol. The natural numbers less than 100 are $1,2,3,\ldots,99$. The collection is well defined.

∴ It is a set.

(vi) A collection of novels written by the writer Munshi Prem Chand.

Sol. For any novel we can decide definitely whether it was written by Munshi Prem Chand or not. The collection is well defined.

∴ It is a set.

(vii) The collection of all even integers.

Sol. For any integer we can decide definitely whether it is even or not. The collection is well defined.

∴ It is a set.

(viii) The collection of questions in this Chapter.

Sol. For any question we can decide definitely whether it is in this Chapter or not. The collection is well defined.

∴ It is a set.

(ix) A collection of most dangerous animals of the world.

Sol. Whether an animal is among the most dangerous animals depends on a person's opinion, so the collection is not well defined.

∴ It is not a set.

(i) $5\ldots A$

(ii) $8\ldots A$

(iii) $0\ldots A$

(iv) $4\ldots A$

(v) $2\ldots A$

(vi) $10\ldots A$

(i) $A=\{x:x$ is an integer and $-3\le x<7\}$

Sol. The integers $x$ with $-3\le x<7$ are $-3,-2,-1,0,1,2,3,4,5,6$ (not $7$).

∴ Roster form of set $A=\{-3,-2,-1,0,1,2,3,4,5,6\}$.

(ii) $B=\{x:x$ is a natural number less $\text{than }6\}$

Sol. The natural numbers less than $6$ are $1,2,3,4,5$.

∴ Roster form of set $B=\{1,2,3,4,5\}$.

(iii) $C=\{x:x$ is a two-digit natural number such that the sum of its digits $\text{is }8\}$

Sol. Let the digits be $a$ (tens) and $b$ (units), with $a\ne0$ and $a+b=8$.

The possible numbers are $17,26,35,44,53,62,71,80$.

∴ Roster form of set $C=\{17,26,35,44,53,62,71,80\}$.

(iv) $D=\{x:x$ is a prime number which is divisor $\text{of }60\}$

Sol. The divisors of $60$ are $1,2,3,4,5,6,10,12,15,20,30,60$.

Among them, the prime numbers are $2,3,5$.

∴ Roster form of set $D=\{2,3,5\}$.

(v) $E=$ The set of all letters in the word TRIGONOMETRY

Sol. In the word TRIGONOMETRY, the letters T, R and O are repeated. Dropping the repetitions, the distinct letters are T, R, I, G, O, N, M, E, Y.

∴ Roster form of set $E=\{T,R,I,G,O,N,M,E,Y\}$.

(vi) $F=$ The set of all letters in the word BETTER

Sol. In the word BETTER, the letters E and T are repeated. Dropping the repetitions, the distinct letters are B, E, T, R.

∴ Roster form of set $F=\{B,E,T,R\}$.

(i) $\{3,6,9,12\}$

(ii) $\{2,4,8,16,32\}$

(iii) $\{5,25,125,625\}$

(iv) $\{2,4,6,\ldots\}$

(v) $\{1,4,9,\ldots,100\}$

Therefore,$\{2,4,6,\ldots\}=\{x:x$ is an even natural $\text{number}\}$.

(i) $A=\{x:x\text{ is an odd natural number}\}$

Sol. The odd natural numbers are $1,3,5,7,\ldots$

∴ $A=\{1,3,5,7,\ldots\}$

(ii) $B=\{x:x\text{ is an integer},\ -\frac12<x<\frac92\}$

Sol. $-\frac12<x<\frac92\Rightarrow-0.5<x<4.5$. The integers in this range are $0,1,2,3,4$.

∴ $B=\{0,1,2,3,4\}$

(iii) $C=\{x:x\text{ is an integer},\ x^2\le4\}$

Sol. $x^2\le4\Rightarrow |x|\le2\Rightarrow-2\le x\le2$. The integers in this range are $-2,-1,0,1,2$.

∴ $C=\{-2,-1,0,1,2\}$

(iv) $D=\{x:x$ is a letter in the word $\text{“LOYAL”}\}$

Sol. The distinct letters in the word LOYAL are L, O, Y, A.

∴ $D=\{L,O,Y,A\}$

(v) $E=\{x:x$ is a month of a year not having 31 $\text{days}\}$

Sol. The months not having 31 days are February, April, June, September and November.

∴ $E=\{$February, April, June, September, $\text{November}\}$

(vi) $F=\{x:x$ is a consonant in the English alphabet which $\text{precedes }k\}$.

Sol. The consonants which come before $k$ in the English alphabet are $b,c,d,f,g,h,j$.

∴ $F=\{b,c,d,f,g,h,j\}$

(i) $\{1,2,3,6\}$

(ii) $\{2,3\}$

(iii) $\{M,A,T,H,E,I,C,S\}$

(iv) $\{1,3,5,7,9\}$

(a) $\{x:x$ is a prime number and a divisor $\text{of }6\}$

(b) $\{x:x$ is an odd natural number less $\text{than }10\}$

(c) $\{x:x$ is natural number and divisor $\text{of }6\}$

(d) $\{x:x$ is a letter of the word $\text{MATHEMATICS}\}$.