M.C.Q (1 Marks)

Question 511 Mark

Let $f: R \rightarrow R$ be defined by $f(x)=x+|x|$. Then $f(x)$ is

Answer

(d) : Given, $f(x)=x+|x|$
Now, $f(-2)=-2+|-2|=-2+2=0$
and $f(-3)=-3+|-3|=-3+3=0$
Hence, $f$ is not one-one
Also, $f(x)=\left\{\begin{array}{ll}x+x & \text { if } x \geq 0 \\ x-x & \text { if } x<0\end{array} \Rightarrow f(x)=\left\{\begin{array}{ll}2 x, & x \geq 0 \\ 0, & x<0\end{array}\right.\right.$
Thus, $f(x)=2 x \geq 0$ for all $x \geq 0$ and $f(x)=0$ for $x<0$. This means that $f(x)$ cannot be negative for any $x \in R$. So, $f$ is not onto. Note that $R_f=[0, \infty)$, which is a proper subset of $R$.

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Question 521 Mark

Let $R$ be a relation on the set $N$ be defined by $\{(x, y): x, y \in N, 2 x+y=41\}$. Then, $R$ is

Answer

(d) $: R=\{(x, y): x, y \in N, 2 x+y=41\}$
Reflexive : $(1,1) \notin R$ as $2 \cdot 1+1=3 \neq 41$. So, $R$ is not reflexive.
Symmetric : $(1,39) \in R$ but $(39,1) \notin R$. So $R$ is not symmetric.
Transitive : $(20,1) \in R$ and $(1,39) \in R$. But $(20,39) \notin R$, so $R$ is not transitive.

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Question 531 Mark

If $A$ and $B$ are finite sets containing respectively $m$ and $n$ elements, then find the number of relation that can be defined from $A$ to $B$.

Answer

(b) : Let $R$ be a relation from $A$ to $B$, then $R \subset A \times B$. This means that the number of relations from $A$ to $B$ is equal to the number of subsets of $A \times B$.
Now, $O(A)=m$ and $O(B)=n$
$\Rightarrow O(A \times B)=m n$
$\therefore \quad$ Number of subsets of $A \times B=2^{m n}$
$\left(\because O(\right.$ Power set of $\left.A)=2^{O(A)}\right)$
$\therefore \quad$ Number of relations from $A$ to $B=2^{m n}$

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Question 541 Mark

Let $A=\{a, b, c\}$ and let $R=\{(a, a),(a, b)$, $(b, a)\}$. Then, $R$ is

Answer

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Question 551 Mark

Let $A=\{1,2,3,4\}$ and $R$ be a relation in $A$ given by $R=\{(1,1),(2,2),(3,3),(4,4),(1,2)$, $(2,1),(3,1)\}$. Then, $R$ is

Answer

(a) : Reflexive: $(1,1),(2,2),(3,3),(4,4) \in R$;
$\therefore \quad R$ is reflexive.

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MCQ 561 Mark

If $A=\{1,2,3\}$ then the number of equivalence relations with element (1, 2) is-