Exercise 1.3 - Chapter 1 Set Language 9th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

$\operatorname{Ex} 1.3$
Question 1.

Using the given venn diagram, write the elements of
(i) $\mathrm{A}$
(ii) $\mathrm{B}$
(iii) $\mathrm{A} \cup \mathrm{B}$
(iv) $A \cap B$
(v) $A-B$
(vi) $B-A$
(vii) $\mathrm{A}^{+}$
(viii) $\mathrm{B}^{+}$
(ix) U

Solution:
(i) $\mathrm{A}=\{2,4,7,8,10\}$
(ii) $\mathrm{B}=\{3,4,6,7,9,11\}$
(iii) $\mathrm{A} \cup \mathrm{B}=\{2,3,4,6,7,8,9,10,11\}$
(iv) $\mathrm{A} \cap \mathrm{B}=\{4,7\}$
(v) $\mathrm{A}-\mathrm{B}=\{2,8,10\}$.
(vi) $\mathrm{B}-\mathrm{A}=\{3,6,9,11\}$
(vii) $\mathrm{A}^{+}=\{1,3,6,9,11,12\}$
(viii) $\mathrm{B}^{\top}=\{1,2,8,10,12\}$
(ix) $\mathrm{U}=\{1,2,3,4,6,7,8,9,10,11,12\}$.

Question $2 .$
Find $\mathrm{A} \cup \mathrm{B}, \mathrm{A} \cap \mathrm{B}, \mathrm{A}-\mathrm{B}$ and $\mathrm{B}-\mathrm{A}$ for the following sets.
(i) $\mathrm{A}=\{2,6,10,14\}$ and $\mathrm{B}=\{2,5,14,16\}$
(ii) $\mathrm{A}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{c}, \mathrm{u}\}$ and $\mathrm{B}=\{\mathrm{a}, \mathrm{c}, \mathrm{i}, \mathrm{o}, \mathrm{u}\}$
(iii) $\mathrm{A}=\{\mathrm{x}: \mathrm{x} \in \mathrm{N}, \mathrm{x} \leq 10\}$ and $\mathrm{B}=\{\mathrm{x}: \mathrm{x} \in \mathrm{W}, \mathrm{x}<6\}$
(iv) $\mathrm{A}=$ Set of all letters in the word "mathematics" and $\mathrm{B}=$ Set of all letters in the word "geometry"
Solution:

(i) $\mathrm{A}=\{2,6,10,14\}$ and $\mathrm{B}=\{2,5,14,16\}$
$\mathrm{A} \cup \mathrm{B}=\{2,6,10,14\} \cup\{2,5,14,16\}=\{2,5,6,10,14,16\}$
$\mathrm{A} \cap \mathrm{B}=\{2,6,10,14\} \cap\{2,5,14,16\}=\{2,14\}$
$A-B=\{2,6,10,14\}-\{2,5,14,16\}=\{6,10\}$
$B-A=\{2,5,14,16\}-\{2,6,10,14\}=\{5,16\}$
(ii) $\mathrm{A}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{c}, \mathrm{u}\}$ and $\mathrm{B}=\{\mathrm{a}, \mathrm{c}, \mathrm{i}, \mathrm{o}, \mathrm{u}\}$
$A \cup B=\{a, b, c, c, u) \cup\{a, c, i, o, u)=\{a, b, c, c, i, o, u\}$
$\mathrm{A} \cap \mathrm{B}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{c}, \mathrm{u}\} \cap\{\mathrm{a}, \mathrm{c}, \mathrm{i}, \mathrm{o}, \mathrm{u}\}\{\mathrm{a}, \mathrm{c}, \mathrm{u}\}$
$A-B=\{a, b, c, c, u)-\{a, c, i, o, u\}=\{b, c\}$
$B-A=\{a, c, i, o, u\}-\{a, b, c, c, u\}=\{i, o\}$
(iii) $x \in\{1,2,3, \ldots \ldots \ldots\} ; x \in\{0,1,2,3,4,5, \ldots \ldots \ldots\}$
$A=\{1,2,3,4,5,6,7,8,9,10\}$
$\mathrm{B}=\{0,1,2,3,4,5\}$
$\mathrm{A} \cup \mathrm{B}=\{1,2,3,4,5,6,7,8,9,10\} \cup\{0,1,2,3,4,5\}=\{0,1,2,3,4,5,6,7,8,9,10\}$
$\mathrm{A} \cap \mathrm{B}=\{1,2,3,4,5,6,7,8,9,10\} \cap\{0,1,2,3,4,5\}=\{1,2,3,4,5\}$
$\mathrm{A}-\mathrm{B}=\{1,2,3,4,5,6,7,8,9,10\}-\{0,1,2,3,4,5\}=\{6,7,8,9,10\}$
$\mathrm{B}-\mathrm{A}=\{0,1,2,3,4,5\}-\{1,2,3,4,5,6,7,8,9,10\}=\{0\}$
(iv) $\mathrm{A}=\{\mathrm{m}, \mathrm{a}, \mathrm{t}, \mathrm{h}, \mathrm{c}, \mathrm{i}, \mathrm{c}, \mathrm{s}), \mathrm{B}=\{\mathrm{g}, \mathrm{c}, \mathrm{o}, \mathrm{m}, \mathrm{t}, \mathrm{r}, \mathrm{y})$
$A \cup B=\{m, a, t, h, c, i, c, s\} \cup\{g, c, o, m, t, r, y\}=\{m, a, t, h, c, i, c, s, g, c, r, y)$
$A \cap B=\{m, a, t, h, c, i, c, s\} \cap\{g, c, o, m, t, r, y\}=\{m, t, c\}$
$A-B=\{m, a, t, h, c, i, c, s\} \cup\{g, c, o, m, t, r, y\}=\{a, h, i, c, s)$
$\mathrm{B}-\mathrm{A}=\{\mathrm{m}, \mathrm{a}, \mathrm{t}, h, \mathrm{c}, \mathrm{i}, \mathrm{c}, 5\} \cap\{\mathrm{g}, \mathrm{c}, \mathrm{o}, \mathrm{m}, \mathrm{t}, \mathrm{r}, \mathrm{y}\}=\{\mathrm{g}, \mathrm{o}, \mathrm{r}, \mathrm{y}\}$

Question $3 .$
If $U=\{a, b, c, d, c, f g, h\}, A=\{b, d, f, h\}$ and $B=\{a, d, c, h\}$, find the following sets.
(i) $\mathrm{A}^{\prime}$
(ii) $\mathrm{B}^{+}$
(iii) $A^{+} \cup B^{*}$
(iv) $\mathrm{A}^{+} \cap \mathrm{B}^{*}$
(v) $(A \cup B)^{*}$
(vi) $(A \cap B)^{+}$
(vii) $\left(A^{*}\right)^{+}$
(viii) $\left(B^{+}\right)^{+}$
Solution:
(i) $A^{\prime}=U-A=\{a, b, c, d, c, f, g, y\}-\{b, d, f, h\}=\{a, c, c, g\}$
(ii) $B^{+}=U-B=\{a, b, c, d, c, f, g, y)-\{a, d, c, h]=\{b, c, f, g\}$
(iii) $\mathrm{A}^{\prime} \cup \mathrm{B}^{\prime}=\{\mathrm{a}, \mathrm{c}, \mathrm{c}, \mathrm{g}\} \cup\{\mathrm{b}, \mathrm{c}, \mathrm{f}, \mathrm{g}\}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{c}, \mathrm{fg}\}$
(iv) $A^{\prime} \cap B^{\prime}=\{a, c, c, g\} \cap\{b, c, f, g\}=\{c, g\}$
(v) $(A \cup B)^{+}=U-(A \cup B)=\{a, b, c, d, c, f, g, y)-\{a, b, d, c, f, h\}=\{c, g\}$
(vi) $(A \cap B)^{+}=U-(A \cap B)=\{a, b, c, d, c, f, g, y\}-\{d, h\}=\{a, b, c, c, f, g\}$
(vii) $\left(A^{+}\right)^{+}=U-A^{+}=\{a, b, c, d, e, f, g, h\}-\{a, c, c, g\}=\{b, d, f, h)$
(viii) $\left(\mathrm{B}^{\prime}\right)^{+}=\mathrm{U}-\mathrm{B}^{+}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{c}, \mathrm{f}, \mathrm{g}, \mathrm{h}\}-\{\mathrm{b}, \mathrm{c}, \mathrm{f}, \mathrm{g}\}=\{\mathrm{a}, \mathrm{d}, \mathrm{c}, \mathrm{h}\}$
 

Question $4 .$
Let $\mathrm{U}=\{0,1,2,3,4,5,6,7\}, \mathrm{A}=\{1,3,5,7\}$ and $\mathrm{B}=\{0,2,3,5,7\}$, find the following sets.
(i) $\mathrm{A}^{\prime}$
(ii) $\mathrm{B}^{+}$
(iii) $\mathrm{A} \cdot \cup \mathrm{B}$ *
(iv) $A^{\prime} \cap B^{\prime}$
(v) $(A \cup B)^{+}$
(vi) $(A \cap B)^{+}$
(vii) $\left(A^{+}\right)^{+}$
(viii) $\left(\mathrm{B}^{\prime}\right)^{+}$
Solution:
(i) $\mathrm{A}^{*}=\mathrm{U}-\mathrm{A}=\{0,1,2, \mathrm{y}, 4,5,6,7\}-\{1,3,5,7\}=\{0,2,4,6\}$
(ii) $\mathrm{B}^{\prime}=\mathrm{U}-\mathrm{B}=\{0,1,2,3,4,5,6,7\}-\{0,2,3,5,7\}=\{1,4,6\}$
(iii) $\mathrm{A}^{+} \cup \mathrm{B}^{\prime}=\{0,2,4,6\} \cup\{1,4,6\}=\{0,1,2,4,6\}$
(iv) $\mathrm{A}^{+} \cap \mathrm{B}^{\prime}=\{0,2,4,6\} \cap\{1,4,6\}=\{4,6\}$
(v) $(\mathrm{A} \cup \mathrm{B})^{+}=\mathrm{U}-(\mathrm{A} \cup \mathrm{B})=\{0,1,2,3,4,5,6,7\}-\{0,1,2,3,5,7\}=\{4,6\}$
(vi) $(A \cap B)^{*}=U-(A \cap B)=\{0,1,2,3,4,5,6,7\}-\{3,5,7\}=\{0,1,2,4,6\}$
(vii) $\left(\mathrm{A}^{+}\right)^{+}=\mathrm{U}-\mathrm{A}^{+}=\{0,1,2,3,4,5,6,7\}-\{0,2,4,6\}=\{1,3,5,7\}$
(viii) $\left(\mathrm{B}^{\prime}\right)^{+}=\mathrm{U}-\mathrm{B}^{+}=\{0,1,2,3,4,5,6,7\}-\{1,4,6\}=\{0,2,3,5,7\}$.
 

Question $5 .$
Find the symmetric difference between the following sets.
(i) $\mathrm{P}=\{2,3,5,7,11\}$ and $\mathrm{Q}=\{1,3,5,11\}$
(ii) $\mathrm{R}=\{1, \mathrm{~m}, \mathrm{n}, \mathrm{o}, \mathrm{p}\}$ and $\mathrm{S}=\{\mathrm{j}, \mathrm{l}, \mathrm{n}, \mathrm{q})$
(iii) $X=\{5,6,7\}$ and $Y=\{5,7,9,10\}$

Solution:

(i) $\mathrm{P}=\{2,3,5,7,11\}$
$\begin{aligned}
&Q=\{1,3,5,11\} \\
&P-Q=\{2,3,5,7,11\}-\{1,3,5,11\}=\{2,7\} \\
&Q-P=\{1,3,5,11\}-\{2,3,5,7,11\}=\{1\} \\
&P \Delta Q=(P-Q) \cup(Q-P)=\{2,7\} \cup\{1\}=\{1,2,7\}
\end{aligned}$
(ii) $\mathrm{R}=\{1, \mathrm{~m}, \mathrm{n}, \mathrm{o}, \mathrm{p}\}$
$\begin{aligned}
&\mathrm{S}=\{\mathrm{j}, 1, \mathrm{n}, \mathrm{q}\} \\
&\mathrm{R}-\mathrm{S}=\{1, \mathrm{~m}, \mathrm{n}, \mathrm{o}, \mathrm{p})-\{\mathrm{j}, 1, \mathrm{n}, \mathrm{q}\}=\{\mathrm{m}, \mathrm{o}, \mathrm{p}) \\
&\mathrm{s}-\mathrm{R}=\{\mathrm{j}, \mathrm{l}, \mathrm{n}, \mathrm{q})-\{\mathrm{l}, \mathrm{m}, \mathrm{n}, \mathrm{o}, \mathrm{p}\}=\{\mathrm{j}, \mathrm{q}\} \\
&\mathrm{R} \Delta \mathrm{S}=(\mathrm{R}-\mathrm{S}) \cup(\mathrm{S}-\mathrm{R})=\{\mathrm{m}, \mathrm{o}, \mathrm{p}) \cup\{\mathrm{j}, \mathrm{q}\}=\{\mathrm{j}, \mathrm{m}, \mathrm{o}, \mathrm{p}, \mathrm{q})
\end{aligned}$
$\begin{aligned}
&\text { (iii) } X=\{5,6,7\} \\
&Y=\{5,7,9,10\} \\
&X-Y=\{5,6,7\}-\{5,7,9,10\}-\{6\} \\
&Y-X=\{5,6,9,10\}-\{5,6,7\}=\{9,10\} \\
&X \Delta Y=(X-Y) \cup(Y-X)=\{6\} \cup\{9,10\}=\{6,9,10\}
\end{aligned}$

Question 6.
Using the set symbols, write down the expressions for the shaded region in the following

(i)

Solution:
(i) $X-Y$
(ii) $(X \cup Y)^{+}$
(iii) $(X-Y) \cup(X-Y)$

Question $7 .$
Let A and B be two overlapping sets and the universal set U. Draw appropriate Venn diagram for each of the following,
(i) $\mathrm{A} \cup \mathrm{B}$
(ii) $\mathrm{A} \cap \mathrm{B}$
(iii) $(A \cap B)^{'}$
(iv) $(B-A)^{'}$
(v) $\mathrm{A}^{'} \cup \mathrm{B}^{'}$
(vi) $\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}$
(vii)What do you observe from the diagram (iii) and (v)?
Solution