If $(x, y) \in N \times N$, then $x y=x^2$ is a relation that is
A
Symmetric
B
Reflexive
C
Transitive
✓
Equivalence
Answer
Correct option: D.
Equivalence
(D) Equivalence Hint: Let $x \in R$, then $x x=x^2$ $\therefore x$ is related to $x$. $\therefore$ Given relation is reflexive. Let $x=0$ and $y=2$, then $x y=0 \times 2=0=x^2$ $\therefore x$ is related to $y$. Consider, $yx =2 \times 0=0 \neq y ^2$ $\therefore y$ is not related to $x$. $\therefore$ Given relation is not symmetric. Let $x$ be related to $y$ and $y$ be related to $z$. $\therefore xy = x ^2$ and $yz = y ^2$ $\therefore x =\frac{x^2}{y}$ and $z =\frac{y^2}{y}= y$.....[if $y \neq 0$ ] Consider, $x z=\frac{x^2}{y} \times y=x^2$ $\therefore x$ is related to $z$. $\therefore$ Given relation is transitive.
The relation " $>$ " in the set of $N$ (Natural number) is
A
Symmetric
B
Reflexive
✓
Transitive
D
Equivalent relation
Answer
Correct option: C.
Transitive
(C) Transitive Hint: For any $a \in N, a>/ a$ $ \therefore(a, a) \notin R $ $\therefore>$ is not reflexive. For any $a, b \in N$, if $a>b$, then $b>a$. $\therefore>$ is not symmetric. For any $a, b, c \in N$, if $a>b$ and $b>c$, then $a>c$ $\therefore>$ is transitive.
In a city $20 \%$ of the population travels by car, $50 \%$ travels by bus and $10 \%$ travels by both car and bus. Then, persons travelling by car or bus are
A
$80 \%$
B
$40 \%$
✓
$60 \%$
D
$70 \%$
Answer
Correct option: C.
$60 \%$
(C) $60 \%$
Hint:
Let $C=$ Population travels by car
$B =$ Population travels by bus
$n(C)=20 \%, n(B)=50 \%, n(C \cap B)=10 \% $
$n(C \cup B)=n(C)+n(B)-n(C \cap B)$
$=20 \%+50 \%-10 \%$
$=60 \%$