Question

Let $R_1$ be a relation from $A=\{1,3,5,7\}$ to $B=\{2,4,6,8\}$ and $R_2$ be another relation from $B$ to $C=\{1,2,3,4\}$ as defined below:

(i) An element $x$ in $A$ is related to an element $y$ in $B$ (under $R_1$) if $x+y$ is divisible by 3.

(ii) An element $x$ in $B$ is related to an element $y$ in $C$ (under $R_2$) if $x+y$ is even but not divisible by 3.

Which is the composite relation $R_1{R}_2$ from$A$ to $C$ ?

• A. $R_1{R}_2=\left\{\right(1,2),(1,4),(3,3),(5,4),(7,3\left)\right\}$
• B. $R_1{R}_2=\left\{\right(1,2),(1,3),(3,2),(5,2),(7,3\left)\right\}$
• C. $R_1{R}_2=\left\{\right(1,2),(3,2),(3,4),(5,4),(7,2\left)\right\}$
• D. $R_1{R}_2=\left\{\right(3,2),(3,4),(5,1),(5,3),(7,1\left)\right\}$

Answer 1

The correct option is C $R_1{R}_2=\left\{\right(1,2),(3,2),(3,4),(5,4),(7,2\left)\right\}$

$R_1=\left\{\text{(An element x in A, An element y in B) if x+y divisible by 3}\right\}$
$R_1=\left\{\right(1,2),(1,8),(3,6),(5,4),(7,2),(7,8\left)\right\}$
$R_2=\left\{\text{(An element x in B, An element y in C) if x + y is even but not divisible by 3}\right\}$
$R_2=\left\{\right(2,2),(4,4),(6,2),(6,4),(8,2\left)\right\}$
So, $R_1\circ R_2=\left\{\right(1,2),(3,2),(3,4),(5,4),(7,2\left)\right\}$