Embibe Experts Solutions for Chapter: Set Theory and Relations, Exercise 1: Exercise-1

If $\left(2x+y,7\right)=\left(5,y-3\right)$, then find $x$ and $y$.

If $A\times B=\left\{\left(1,2\right), \left(1,3\right), \left(1,6\right), \left(7,2\right), \left(7,3\right), \left(7,6\right)\right\}$, then find sets $A$ and $B$.

If $A=\left\{\mathrm{x},\mathrm{y},\mathrm{z}\right\}$ and $B=\left\{1,2\right\}$, then find the number of relations from $A$ to $B$.

Write $R=\left\{\left(4x+3,1-x\right) : x\le 2, x\in N\right\}$ in set builder form

Let $\mathrm{L}$ be the set of all lines in a plane and let $\mathrm{R}$ be a relation defined on $\mathrm{L}$ by the rule $\left(\mathrm{x}, \mathrm{y}\right)\in \mathrm{R} \Rightarrow \mathrm{x}$ is perpendicular to $\mathrm{y}$. Then prove that $\mathrm{R}$ is a symmetric relation on $\mathrm{L}$.

Let $\mathrm{R}$ be a relation on the set of all lines in a plane defined by $\left(l_1, l_2\right)\in \mathrm{R}\Rightarrow$ line $l_1$ is parallel to line $l_2$. Prove that $\mathrm{R}$ is an equivalence relation.

Let $R=\left\{\right(x,y):x,y\in A,x+y=5\}$ where $A=\{1,2,3,4,5\}$ then $R$ is
 

Let $S$ be a set of all square matrices of order $2 .$ If a relation $\mathrm{R}$ defined on set $S$ such that $\mathrm{ARB}\Rightarrow \mathrm{AB}=\mathrm{O},$ where $\mathrm{O}$ is zero square matirx of order $2,$ then relation $\mathrm{R}$ is $(\mathrm{A},\mathrm{B}\in \mathrm{S})$