Exercise 1.2 - Chapter 1 Set Language 9th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Ex $1.2$
Question $1 .$
Find the cardinal number of the following sets.
(i) $\mathrm{M}=\{\mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{s}, \mathrm{t}, \mathrm{u}\}$
(ii) $\mathrm{P}=\{\mathrm{x}: \mathrm{x}=3 \mathrm{n}+2, \mathrm{n} \in \mathrm{W}$ and $\mathrm{x}<15\}$
(iii) $\mathrm{Q}=\left\{\mathrm{v}: \mathrm{v}=\frac{4}{3 n}, \mathrm{n} \in \mathrm{N}\right.$ and $\left.2<\mathrm{n} \leq 5\right\}$
(iv) $R=\{x: x$ is an integers, $x \in Z$ and $-5 \leq x<5\}$
(v) $\mathrm{S}=$ The set of all leap years between 1882 and 1906 .
Solution:
(i) $n(M)=6$
(ii) $\mathrm{W}=\{0,1,2,3, \ldots \ldots\}$
if $\mathrm{n}=0, \mathrm{x}=3(0)+2=2$
if $n=1, x=3(1)+2=5$
if $n=2, x=3(2)+2=8$
if $n=3, x=3(3)+2=11$
if $\mathrm{n}=4, \mathrm{x}=3(4)+2=14$
$\therefore \mathrm{P}=\{2,5,8,11,14\}$
$n(P)=5$
(iii) $\mathrm{N}=\{1,2,3,4, \ldots . .\}$
$\mathrm{n} \in\{3,4,5\}$
if $n=3, \quad y=\frac{4}{3(3)}=\frac{4}{9}$
if $n=4, \quad y=\frac{4}{3(4)}=\frac{4}{12}$
if $n=5, \quad y=\frac{4}{3(5)}=\frac{4}{15}$
$Q=\left\{\frac{4}{9}, \frac{4}{12}, \frac{4}{15}\right\}$
$n(Q)=3$

$\begin{aligned}
&\text { (iv) } x \in z \\
&R=\{-5,-4,-3,-2,-1,0,1,2,3,4\} \\
&n(R)=10 \\
&\text { (v) } S=\{1884,1888,1892,1896,1904\} \\
&n(S)=5
\end{aligned}$
 

Question $2 .$
Identify the following sets as finite or infinite.
(i) $\mathrm{X}=$ The set of all districts in Tamilnadu.
(ii) $Y=$ The set of all straight lines passing through a point.
(iii) $A=\{x: x \in Z$ and $x<5\}$
(iv) $\mathrm{B}=\left\{\mathrm{x}: \mathrm{x}^{2}-5 \mathrm{x}+6=0, \mathrm{x} \in \mathrm{N}\right\}$
Solution:
(i) Finite set
(ii) Infinite set
(iii) $\mathrm{A}=\{\ldots \ldots,-2,-1,0,1,2,3,4\}$
$\therefore$ Infinite set
(iv) $x^{2}-5 x+6=0$ $(x-3)(x-2)=0$ $B=\{3,2\}$ $\therefore$ Finite set.
 

Question $3 .$
Which of the following sets are equivalent or unequal or equal sets?
(i) $\mathrm{A}=$ The set of vowels in the English alphabets.
$\mathrm{B}=$ The set of all letters in the word "VOWEL"
(ii) $\mathrm{C}=\{2,3,4,5\}$
$\mathrm{D}=\{\mathrm{x}: \mathrm{x} \in \mathrm{W}, 1<\mathrm{x}<5\}$
(iii) $\mathrm{X}=\mathrm{A}=\{\mathrm{x}: \mathrm{x}$ is a letter in the word "LIFE" $\}$
$\mathrm{Y}=\{\mathrm{F}, \mathrm{I}, \mathrm{L}, \mathrm{E}\}$
(iv) $\mathrm{G}=\{\mathrm{x}: \mathrm{x}$ is a prime number and $3<\mathrm{x}<23\}$
$\mathrm{H}=\{\mathrm{x}: \mathrm{x}$ is a divisor of 18$\}$
Solution:
(i) $\mathrm{A}=\{\mathrm{a}, \mathrm{e}, \mathrm{i}, \mathrm{o}, \mathrm{u}\}$
$\mathrm{B}=\{\mathrm{V}, \mathrm{O}, \mathrm{W}, \mathrm{E}, \mathrm{L}\}$
The sets $\mathrm{A}$ and $\mathrm{B}$ contain the same number of elements.
$\therefore$ Equivalent sets

$\begin{aligned}
&\text { (ii) } \mathrm{C}=\{2,3,4,5\} \\
&\mathrm{D}=\{2,3,4\}
\end{aligned}$
$\therefore$ Unequal sets
(iii) $\mathrm{X}=\{\mathrm{L}, \mathrm{I}, \mathrm{F}, \mathrm{E}\}$
$\mathrm{Y}=\{\mathrm{F}, \mathrm{I}, \mathrm{L}, \mathrm{E}\}$
The sets $\mathrm{X}$ and $\mathrm{Y}$ contain the exactly the same elements.
$\therefore$ Equal sets.
(iv) $\mathrm{G}=\{5,7,11,13,17,19\}$
$\mathrm{H}=\{1,2,3,6,9,18\}$
$\therefore$ Equivalent sets.

Question $4 .$
Identify the following sets as null set or singleton set.
(i) $\mathrm{A}=(\mathrm{x}: \mathrm{x} \in \mathrm{N}, 1<\mathrm{x}<2\}$
(ii) $\mathrm{B}=$ The set of all even natural numbers which are not divisible by 2 .
(iii) $\mathrm{C}=\{0\}$
(iv) $\mathrm{D}=$ The set of all triangles having four sides.
Solution:
(i) $A=\{\} \because$ There is no element in between 1 and 2 in Natural numbers.
$\therefore$ Null set
(ii) $\mathrm{B}=\{\} \because \mathrm{All}$ even natural numbers are divisible by 2 .
$\therefore B$ is Null set
(iii) $\mathrm{C}=\{0\}$
$\therefore$ Singleton set
(iv) $\mathrm{D}=\{\}$
$\because$ No triangle has four sides.

Question $5 .$
State which pairs of sets are disjoint or overlapping?
(i) $A=\{f, i, a, s\}$ and $B=\{a, n, f, h, s)$
(ii) $\mathrm{C}=\{\mathrm{x}: \mathrm{x}$ is a prime number, $\mathrm{x}>2\}$ and $\mathrm{D}=\{\mathrm{x}: \mathrm{x}$ is an even prime number $\}$
(iii) $\mathrm{E}=\{\mathrm{x}: \mathrm{x}$ is a factor of 24$\}$ and $\mathrm{F}=\{\mathrm{x}: \mathrm{x}$ is a multiple of $3, \mathrm{x}<30\}$ Solution:
(i) $A=\{f, i, a, s\}$
$B=\{a, n, f, h, s\}$
$A \cap B=\{f, i, a, s\} \cap\{a, n, f h, s\}=\{f, a, s\}$
Since $A \cap B \neq \phi, A$ and $B$ are overlapping sets.
(ii) $\mathrm{C}=\{3,5,7,11, \ldots \ldots\}$
$\mathrm{D}=\{2\}$
$\mathrm{C} \cap \mathrm{D}=\{3,5,7,11, \ldots \ldots\} \cap\{2\}=\{\}$
Since $C \cap D=\varnothing, C$ and $D$ are disjoint sets.
(iii) $\mathrm{E}=\{1,2,3,4,6,8,12,24\}$
$\mathrm{F}=\{3,6,9,12,15,18,21,24,27\}$
$\mathrm{E} \cap \mathrm{F}=\{1,2,3,4,6,8,12,24\} \cap\{3,6,9,12,15,18,21,24,27\}$ $=\{3,6,12,24\}$
Since $\mathrm{E} \cap \mathrm{F} \neq \phi, \mathrm{E}$ and $\mathrm{F}$ are overlapping sets.

Question 6.
If $\mathrm{S}=\{$ square,rectangle,circle,rhombus,triangle $\}$, list the elements of the following subset of $S$.
(i) The set of shapes which have 4 equal sides.
(ii) The set of shapes which have radius.
(iii) The set of shapes in which the sum of all interior angles is $180^{\circ}$
(iv) The set of shapes which have 5 sides.
Solution:
(i) Square, Rhombus
(ii) Circle
(iii) Triangle
(iv) Null set.
 

Question $7 .$
If $A=\{a,\{a, b\}\}$, write all the subsets of $A$.
Solution:
$A=\{a,\{a, b\}\}$ subsets of $A$ are \{\}$\{a\},\{a, b\},\{a,\{a, b\}\}$
 

Question $8 .$
Write down the power set of the following sets.
(i) $A=\{a, b\}$
(ii) $\mathrm{B}=\{1,2,3\}$
(iii) $\mathrm{D}=\{\mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{s}\}$
(iv) $\mathrm{E}=\mathscr{\emptyset}$
Solution:
(i) The subsets of $A$ are $\varnothing,\{a\},\{b\},\{a, b\}$
The power set of $A$
$P(A)=\{D,\{a\},\{b\},\{a, b\}\}$
(ii) The subsets of B are $\phi,\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}$
The power set of B
$P(B)=\{\varnothing,\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}\}$

(iii) The subset of $D$ are $\emptyset,\{p\},\{q\},\{r\},\{s\},\{p, q\},\{p, r\},\{p, s\},\{q, r\},\{q, s\},\{r, s\}$, $\{p, q, r\},\{q, r, s\},\{p, r, s\},\{p, q, s\},\{p, q, r, s\}\}$
The power set of $D$
$P(D)=\{\varnothing,\{p\},\{q\},\{r\},\{s\},\{p, q\},\{p, r\},\{p, s\},\{q, r\},\{q, s\},\{r, s\},\{p, q, r\},\{q, r$ $\mathrm{s}\},\{\mathrm{p}, \mathrm{r}, \mathrm{s}\},\{\mathrm{p}, \mathrm{q}, \mathrm{s}\},\{\mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{s}\}$
(iv) The power set of $E$
$P(E)=\{\} \text {. }$
 

Question $9 .$
Find the number of subsets and the number of proper subsets of the following sets.
(i) $\mathrm{W}=\{$ red,blue, yellow $\}$
(ii) $X=\left\{x^{2}: x \in N, x^{2} \leq 100\right\}$.
Solution:
(i) Given $W=\{$ red, blue, yellow $\}$
Then $\mathrm{n}(\mathrm{W})=3$
The number of subsets $=n[\mathrm{P}(\mathrm{W})]=2^{3}=8$
The number of proper subsets $=n[\mathrm{P}(\mathrm{W})]-1=2^{3}-1=8-1=7$
(ii) Given $\mathrm{X}=\{1,2,3,$,
$X^{2}=\{1,4,9,16,25,36,49,64,81,100\}$
$n(X)=10$
The Number of subsets $=n[\mathrm{P}(\mathrm{X})]=2^{10}=1024$
The Number of proper subsets $=n[\mathrm{P}(\mathrm{X})]-1=2^{10}-1=1024-1=1023$.
 

Question $10 .$
(i) If $n(A)=4$, find $n[P(A)]$.
(ii) If $n(A)=0$, find $n[P(A)]$.
(iii) If $n[\mathrm{P}(\mathrm{A})]=256$, find $n(A)$.
Solution:
(i) $\mathrm{n}(\mathrm{A})=4$
$n[P(A)]=2 n=2^{4}=16$
(ii) $n(A)=0$
$\mathrm{n}[\mathrm{P}(\mathrm{A})]=2^{0}=1$
(iii) $n[\mathrm{P}(\mathrm{A})]=256$

$\begin{aligned}
&\mathrm{n}[\mathrm{P}(\mathrm{A})]=28 \\
&\therefore \mathrm{n}(\mathrm{A})=8
\end{aligned}$