Embibe Experts Solutions for Exercise 6: Assignment

If $f\left(x\right)=x^2-3x+1$, find $x\in R$ such that $f\left(2x\right)=f\left(x\right)$.

Let $R$ be a relation on the set of natural numbers $N$ such that $R=\left\{\left(x,y\right):x+2y=41;x,y\in N\right\}$. Find the domain and range of $R$.

Let $A=\left\{1,2,3,4\right\}$ and $B=\left\{x,y,z\right\}.$ Let $R$ be a relation from $A$ to $B$ defined by $B=\left\{x,y,z\right\}.$$R=\{\left(1,x\right),\left(1,z\right),\left(3,x\right)$, $\left(4,y\right)\}$
Draw arrow diagram of relation $R$.

If $A$ and $B$ are two non-empty sets having $n$ elements in common then prove that $A\times B$ and $B\times A$ have $n^2$ elements in common.

For any three sets $A,B,C$, prove that
(i) $A\times \left(B\cup C\right)=\left(A\times B\right)\cup \left(A\times C\right)$
(ii) $A\times \left(B\cap C\right)=\left(A\times B\right)\cap \left(A\times C\right)$

If $A\subseteq B$, show that $A\times A\subseteq \left(A\times B\right)\cap \left(B\times A\right)$

For any sets $A, B, C, D$, prove that $\left(A\times B\right)\cap \left(C\times D\right)=\left(A\cap C\right)\times \left(B\cap D\right)$

Determine domain and range of the relation $R=\{\left(x,y\right):y=\left|x-1\right|,x\in Z$ and $\left|x\right|\le 3\}$