Exercise 2.1 (Revised) - Chapter 2 - Relations & Functions - Ncert Solutions class 11 - Maths

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Chapter 2: Relations & Functions - NCERT Solutions for Class 11 Maths

Ex 2. 1 Question 1.

If $\left(\frac{x}{3}+1, y-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$, find the values of $x$ and $y$.

Answer.

Here $\left(\frac{x}{3}+1, y-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$
$\Rightarrow \frac{x}{3}+1=\frac{5}{3}$ and $y-\frac{2}{3}=\frac{1}{3}$
$\Rightarrow \frac{x}{3}=\frac{5}{3}-1$ and $y=\frac{1}{3}+\frac{2}{3}$
$\Rightarrow \frac{x}{3}=\frac{2}{3}$ and $y=\frac{3}{3}$
$\Rightarrow x=2$ and $y=1$
Ex 2. 1 Question 2.

If the set $A$ has 3 elements and the set $B=\{3,4,5\}$, then find the number of elements $\operatorname{in}(A \times B)$.

Answer.

Number of elements in set $A=3$ and Number of elements in set $B=3$
$\therefore$ Number of elements in $\mathrm{A} \times \mathrm{B}=3 \times 3=9$

Ex 2. 1 Question 3.

If $G=\{7,8\}$ and $H=\{5,4,2\}$, find $G \times H$ and $H \times G$.

Answer.

Given: $\mathrm{G}=\{7,8\}$ and $\mathrm{H}=\{5,4,2\}$
$
\therefore G \times H=\{(7,5),(7,4),(7,2),(8,5),(8,4),(8,2)\}
$

And $H \times G=\{(5,7),(4,7),(2,7),(5,8),(4,8),(2,8)\}$

Ex 2. 1 Question 4.

State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly:
(i) If $\mathbf{P}=\{m, n\}$ and $\mathbf{Q}=\{n, m\}$, then $\mathbf{P} \times \mathbf{Q}=\{(m, n)(n, m)\}$.
(ii) If $\mathrm{A}$ and $\mathrm{B}$ are non-empty sets, then $\mathrm{A} \times \mathrm{B}$ is a non-empty set of ordered pairs $(x, y)$ such that $x \in \mathrm{A}$ and $y \in \mathrm{B}$.
(iii) If $A=\{1,2\}, B=\{3,4\}$, then $A \times(B \cap \phi)=\phi$

Answer.

(i) Here $\mathrm{P}=\{m, n\}$ and $\mathrm{Q}=\{n, m\}$
Number of elements in set $\mathrm{P}=2$ and Number of elements in set $\mathrm{Q}=2$
$\therefore$ Number of elements in $\mathrm{P} \times \mathrm{Q}=2 \times 2=4$
But $\mathrm{P} \times \mathrm{Q}=\{(m, n),(n, m)\}$ and here number of elements in $\mathrm{P} \times \mathrm{Q}=2$
Therefore, statement is false.
Correct statment is $P \times Q=\{(m, m),(n, n),(n, m),(m, n)\}$
(ii) True
(iii) True

Ex 2. 1 Question 5.

If $A=\{-1,1\}$, find $A \times A \times A$.

Answer.

Here $A=\{-1,1\}$
$
\mathrm{A} \times \mathrm{A}=\{(-1,-1),(-1,1),(1,-1),(1,1)\}
$

Ex 2. 1 Question 6.

$\text {If } \mathrm{A} \times \mathrm{B}=\{(a, x),(a, y),(b, x),(b, y)\} \text {, find } A \text { and } B \text {. }$

Answer.

Given: $\mathrm{A} \times \mathrm{B}=\{(a, x),(a, y),(b, x),(b, y)\}$
$\therefore \mathrm{A}=$ set of first elements $=\{a, b\}$ and $\mathrm{B}=$ set of second elements $=\{x, y\}$
Ex 2. 1 Question 7.

Let $A=\{1,2\}, B=\{1,2,3,4\}, C=\{5,6\}$ and $D=\{5,6,7,8\}$. Verify that:
(i) $A \times(B \cap C)=(A \times B) \cap(A \times C)$
(ii) $A \times C$ is a subset of $B \times D$.

Answer.

Given: $\mathrm{A}=\{1,2\}, \mathrm{B}=\{1,2,3,4\}$,
$
C=\{5,6\} \text { and } D=\{5,6,7,8\}
$
(i) $\mathrm{B} \cap \mathrm{C}=\{1,2,3,4\} \cap\{5,6\}=\phi$
$\therefore \mathrm{A} \times \mathrm{B} \cap \mathrm{C}=\{1,2\} \times \phi=\phi$
$A \times B=\{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4)\}$
$A \times C=\{(1,5),(1,6),(2,5),(2,6)$
$\therefore(\mathrm{A} \times \mathrm{B}) \cap(\mathrm{A} \times \mathrm{C})=\phi$
Therefore, from eq. (i) and (ii), $\mathrm{A} \times \mathrm{B} \cap \mathrm{C}$

$
=(\mathrm{A} \times \mathrm{B}) \cap(\mathrm{A} \times \mathrm{C})
$
(ii) $\mathrm{A} \times \mathrm{C}=\{(1,5),(1,6),(2,5),(2,6)$
$
\begin{aligned}
& B \times D=\{(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8), \\
& (4,5),(4,6),(4,7),(4,8),
\end{aligned}
$

Therefore, it is clear that each element of $\mathrm{A} \times \mathrm{C}$ is present in $\mathrm{B} \times \mathrm{D}$.
$
\therefore A \times C \subset B \times D
$
Ex 2. 1 Question 8.

Let $A=\{1,2\}$ and $B=\{3,4\}$, write $\mathrm{A} \times \mathrm{B}$. How many subsets will $\mathrm{A} \times \mathrm{B}$ have? List them.

Answer.

Given: $\mathrm{A}=\{1,2\}$ and $\mathrm{B}=\{3,4\}$
$
\therefore A \times B=\{(1,3),(1,4),(2,3),(2,4)\}
$

Number of elements in $\mathrm{A} \times \mathrm{B}=4$

Therefore, Number of subsets of $\mathrm{A} \times \mathrm{B}=2^4=16$
$
\phi,\{(2,3)\},\{(1,4)\},\{(2,3)\},\{(2,4)\},\{(1,3),(1,4)\},\{(1,3),(2,3)\},\{(1,3),(2,4)\},\{(1,4),(2,3)\},\{(1,4),(2 \text {, }
$
$4)\},\{(2,3),(2,4)\},\{(1,3),(1,4),(2,3)\},\{(1,3),(1,4),(2,4)\},\{(1,3),(2,3),(2,4)\},\{(1,4),(2,3),(2$, 4) $\},\{(1,3),(1,4),(2,3),(2,4)\}$
Ex 2. 1 Question 9.

Let $\mathrm{A}$ and $\mathrm{B}$ be two sets such that $n(\mathrm{~A})=3$ and $n(\mathrm{~B})=2$. If $(x, 1),(y, 2)(z, 1)$ are in $A \times B$.

Answer.

Here $(x, 1) \in A \times B$
$
\begin{aligned}
& \Rightarrow x \in \mathrm{A} \text { and } 1 \in \mathrm{B} \\
& (y, 2) \in \mathrm{A} \times \mathrm{B}
\end{aligned}
$

$
\begin{aligned}
& \Rightarrow y \in \mathrm{A} \text { and } 2 \in \mathrm{B} \\
& (z, 1) \in \mathrm{A} \times \mathrm{B} \\
& \Rightarrow z \in \mathrm{A} \text { and } 1 \in \mathrm{B}
\end{aligned}
$

But it is given that $n(\mathrm{~A})=3$ and $n(\mathrm{~B})=2$
$
\therefore A=\{x, y, z\} \text { and } B=\{1,2\}
$
Ex 2.1 Question 10.

The Cartesian Product $\mathrm{A} \times \mathrm{A}$ has 9 elements among which are found $(-1,0)$ and $(0$,
1). Find the set $A$ and the remaining elements of $A \times A$.

Answer.

Here $(-1,0) \in A \times A$

$
\begin{aligned}
& \Rightarrow-1 \in \mathrm{A} \text { and } 0 \in \mathrm{A} \\
& (0,1) \in \mathrm{A} \times \mathrm{A} \\
& \Rightarrow 0 \in \mathrm{A} \text { and } 1 \in \mathrm{A} \\
& \therefore-1,0,1 \in \mathrm{A}
\end{aligned}
$

But it is given that $n(A \times A)=9$ which implies that $n(\mathrm{~A})=3$
$
\therefore A=\{-1,0,1\}
$

And $\mathrm{A} \times \mathrm{A}=\{(-1,-1),(-1,0),(-1,1),(0,-1),(0,0),(0,1),(1,-1),(1,0),(1,1)\}$
Therefore, the remaining elements of $\mathrm{A} \times \mathrm{A}$ are
$
(-1,-1),(-1,1),(0,-1),(0,0),(1,-1),(1,0),(1,1)
$