Algebra of Relations

The binary operation $*R\times R\to R,$ is defined as  $a*b=2a+b.$ From the given options choose the value of  $\left(2*3\right)*4$ is equal to

$A=\left\{4,1\right\}$, $B=\left\{1,5\right\}$. Verify commutative property of Cartesian product of sets.

$A=\left\{5,1\right\}$, $B=\left\{1,2\right\}$. Verify commutative property of Cartesian product of sets.

$A=\left\{5,6\right\}$, $B=\left\{6,2\right\}$. Verify commutative property of Cartesian product of sets.

$A=\left\{\mathrm{5},\mathrm{4}\right\}$, $B=\left\{\mathrm{4},3\right\}$. Verify commutative property of Cartesian product of sets.

$A=\left\{1,2\right\}$, $B=\left\{2,3\right\}$. Verify commutative property of Cartesian product of sets.

$A=\left\{p,q\right\}$, $B=\left\{q,r\right\}$ and $C=\left\{p,r\right\}$. Verify associative property of Cartesian product of sets.

$A=\left\{4,3\right\}$, $B=\left\{3,2\right\}$ and $C=\left\{4,2\right\}$. Verify associative property of Cartesian product of sets.

$A=\left\{3,9\right\}$, $B=\left\{9,6\right\}$ and $C=\left\{3,6\right\}$. Verify associative property of Cartesian product of sets.

$A=\left\{1,4\right\}$, $B=\left\{4,5\right\}$ and $C=\left\{1,5\right\}$. Verify associative property of Cartesian product of sets.

$A=\left\{1,2\right\}$, $B=\left\{2,3\right\}$ and $C=\left\{1,3\right\}$. Verify associative property of Cartesian product of sets.

If $A=\left\{p,q\right\}$, $B=\left\{r,s\right\}$ and $C=\left\{q,t\right\}$, then prove that, $A\times (B\cup C)=(A\times B)\cup (A\times C)$.

If $A=\left\{a,b\right\}$, $B=\left\{m,n\right\}$ and $C=\left\{b,y\right\}$, then prove that, $A\times (B\cup C)=(A\times B)\cup (A\times C)$.

If $A=\left\{5,9\right\}$, $B=\left\{6,8\right\}$ and $C=\left\{9,7\right\}$, then prove that, $A\times (B\cup C)=(A\times B)\cup (A\times C)$.

If $A=\left\{b,a\right\}$, $B=\left\{c,e\right\}$ and $C=\left\{a,d\right\}$, then prove that, $A\times (B\cup C)=(A\times B)\cup (A\times C)$.

If $A=\left\{0,1\right\}$, $B=\left\{2,4\right\}$ and $C=\left\{1,3\right\}$, then prove that, $A\times (B\cup C)=(A\times B)\cup (A\times C)$.

The diagram shows the graph of a relation $R$ that maps set $A$ onto set $B$.

Write down $R$ as a set of ordered pairs.

Express the mapping $R$ in the form $x\to ...$