Question

Let $A=\{1,2,3,4\},B=\{1,5,9,11,15,16\}$ and $f=\left\{\right(1,5),(2,9),(3,1),(4,5),(2,11\left)\right\}$ Are the following true?
(i) $f$ is a relation from $A$ to $B$
(ii) $f$ is a function from $A$ to $B$
Justify your answer in each case

Answer 1

$A=\{1,2,3,4\}$ and $B=\{1,5,9,11,15,16\}$
$\therefore A\times B=\left\{\right(1,1),(1,5),(1,9),(1,11),(1,15),(1,16),(2,1),(2,5),(2,9),(2,11),(2,15),(2,16),$

$(3,1),(3,5),(3,9),(3,11),(3,15),(3,16),(4,1),(4,5),(4,9),(4,11),(4,15),(4,16)\}$

It is given that $f=\left\{\right(1,5),(2,9),(3,1),(4,5),(2,11\left)\right\}$
(i) A relation from a non-empty set $A$ to a non-empty set $B$ is a subset of the Cartesian product $A\times B$
It is observed that f is a subset of $A\times B$
Thus $f$ is a relation from $AtoB$.
(ii) Since the element $2$ corresponds to two different images i.e., $9$ and $11$. So, relation $f$ is not a function