Question

Which of the following are relations from the set $A=\{1,2,3,4\}$ to set $B=\{a,b,c\}$?

• A. $\left\{\right(1,a),(1,b),(1,c\left)\right\}$
• B. $\left\{\right(4,a),(4,b),(4,c\left)\right\}$
• C. $\left\{\right(3,a),(3,b),(3,c\left)\right\}$
• D. $\left\{\right(2,a),(2,b),(2,c\left)\right\}$

Answer 1

Answer: (A), (B), (C), (D)

$\left\{\right(2,a),(2,b),(2,c\left)\right\}$

A relation $R$ from a non-empty set $A$ to a non-empty set $B$ is a subset of $A\times B.$
Given that $A=\{1,2,3,4\}$ and $B=\{a,b,c\}.$
$\Rightarrow A\times B=\left\{\right(1,a),(1,b),(1,c),(2,a),(2,b),(2,c),(3,a),(3,b),(3,c),(4,a),(4,b),(4,c\left)\right\}$

Note that $\left\{\right(1,a),(1,b),(1,c\left)\right\}\subseteq A\times B,$

$\left\{\right(2,a),(2,b),(2,c\left)\right\}\subseteq A\times B,$

$\left\{\right(3,a),(3,b),(3,c\left)\right\}\subseteq A\times B$ and

$\left\{\right(4,a),(4,b),(4,c\left)\right\}\subseteq A\times B.$

Thus, they are all relations from $A$ to $B.$