For alpha in N, consider a relation R on N gi | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

For $R$ to be reflexive $\Rightarrow xRx$

$\Rightarrow 3 x +\alpha x =7 x \Rightarrow(3+\alpha) x =7 K$

$\Rightarrow 3+\alpha=7 \lambda \Rightar

Vedclass · Accepted Answer

$4$ is the remainder when $\alpha$ is divided by $7$

Vedclass · Answer

For $R$ to be reflexive $\Rightarrow xRx$

$\Rightarrow 3 x +\alpha x =7 x \Rightarrow(3+\alpha) x =7 K$

$\Rightarrow 3+\alpha=7 \lambda \Rightarrow \alpha=7 \lambda-3=7 N +4, K , \lambda, N \in I$

$	herefore$ when $\alpha$ divided by $7$, remainder is $4$.

$R$ to be symmetric $x R y \Rightarrow y R x$

$3 x +\alpha y =7 N _{1}, 3 y +\alpha x =7 N _{2}$

$\Rightarrow(3+\alpha)( x + y )=7\left( N _{1}+ N _{2}ight)=7 N _{3}$

Which holds when $3+\alpha$ is multiple of $7$

$	herefore \alpha=7 N +4 	ext { (as did earlier) }$

$R$ to be transitive

$xRy \& yRz \Rightarrow xRz$.

$3 x +\alpha y =7 N _{1} \& 3 y +\alpha z =7 N _{2}$

$3 x +\alpha z =7 N _{3}$

$	herefore 3 x +7 N _{2}-3 y =7 N _{3}$

$	herefore 7 N _{1}-\alpha y +7 N _{2}-3 y =7 N _{3}$

$	herefore 7\left( N _{1}+ N _{2}ight)-(3+\alpha) y =7 N _{3}$

$	herefore(3+\alpha) y =7\,N$

Which is true again when $3+\alpha$ divisible by $7$ when $\alpha$ divided by $7$, remainder is $4$ .

For $\alpha \in N$, consider a relation $R$ on $N$ given by $R =\{( x , y ): 3 x +\alpha y$ is a multiple of 7$\}$.The relation $R$ is an equivalence relation if and only if.

Similar Questions

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

Let $R$ and $S$ be two equivalence relations on a set $A$. Then

Let $A = \{p, q, r\}$. Which of the following is an equivalence relation on $A$

Let $R$ be a relation on $Z \times Z$ defined by$ (a, b)$$R(c, d)$ if and only if $ad - bc$ is divisible by $5$ . Then $\mathrm{R}$ is