Set Theory(NCERT Exemplar )

Here we have given the solutions of some important questions of chapter 1 (Set Theory) of NCERT Exemplar book .

Short answer type questions

1. Write the following sets in the roaster form

(i) $\large A=\left \{ x: x\in \mathbb{R}, 2x+11=15 \right \}$

(ii) $\large B=\left \{ x|x^{2} =x, x\in \mathbb{R}\right \}$

(iii) $\large C=\left \{ x|x \, is \, a\, positive \, factor \, of\, a\, prime\, number\, p \right \}$

Sol. (i) On solving the equation $\large 2x+11=15$ , we get

$\large 2x=4 \Rightarrow x=2$

Thus $A=\left \{ 2 \right \}$ .

(ii) $\large x^{2}=x$ gives $\large x(x-1)=0$ which implies $\large x=0, 1$

Thus $B=\left \{ 0,1 \right \}$

(iii) Since every prime no. has only two factors 1 and number itself. This implies

$\large C=\left \{ 1,p \right \}$

2. Write the following sets in the roaster form:

(i) $\large D=\left \{ t|t^{3}=t, t\in \mathbb{R} \right \}$

(ii) $E=\left \{ w|\frac{w-2}{w+3} =3, w\in \mathbb{R}\right \}$

(iii) $F=\left \{ x|x^{4}-5x^{2}+6=0, x\in \mathbb{R} \right \}$

Sol. (i) On solving $t^{3}=t$ , we get $t(t^{2}-1)=0$ , which implies either $t=0$ or $t^{2}-1=0$ .

$\Rightarrow t=-1,0,+1$

Thus $D=\left \{ -1,0,+1 \right \}$

(ii) $\frac{w-2}{w+3}=3$ implies

$w-2=3w+9$

It gives $w=-\frac{11}{2}$ .

Thus $E= \left \{-\frac{11}{2} \right \}$ .

(iii) We have $x^{4}-5x^{2}+6=0$

This implies $x^{4}-3x^{2}-2x^{2}+6=0\Rightarrow (x^{2}-3)(x^{2}-2)=0 \Rightarrow x=\pm \sqrt{3}, \pm \sqrt{2}$ .

Therefore $F=\left \{ -\sqrt{3},-\sqrt{2},\sqrt{2} ,\sqrt{3}\right \}$ .

Q3. If $Y = \left \{ x: x \, is \, a \, positive \, factor \, of \, the \, number\, 2^{p-1}\left(2^{p}-1\right),\, where\, 2^{p} - 1 \, is \, a \, prime \, number\right \}$ . Write $Y$ in the roaster form.

sol. $Y = \left \{ x: x \, is \, a \, positive \, factor \, of \, the \, number\, 2^{p-1}\left(2^{p}-1\right),\, where\, 2^{p} - 1 \, is \, a \, prime \, number\right \}$ .

Since $2^{p}-1$ is prime number , it has only two factors 1 and ( $2^{p}-1$ ) and the factors of $2^{p-1}$ are 1,2,2²,2³,…, $2^{p-1}$ . Therefore $Y= \left\{1,2,2^{2},2^{3},.....,2^{p-1},2^{p}-1\right \}$ .

Q4. $A$ , $B$ and $C$ are subsets of Universal Set $U$ . If $A = \left \{2, 4, 6, 8, 12, 20 \right \}, B= \left \{3,6,9,12,15\right \}, C= \left \{5,10,15,20 \right \}$ and $U$ is the set of all whole numbers, draw a Venn diagram showing the relation of $U, A, B$ and $C$ .