Question

Let $A = \\{ 1,2,3,4\\}$, $B = \\{ 1,5,9,11,15,16\\}$ and $f = \\{ (1,5),(2,9),(3,1),(4,5),(2,11)\\}$
Are the following true?

1. $f$ is the relation from $A{\text{ to }}B$
2. $f$ is the function from $A{\text{ to }}B$
   Justify your answer in each case.

Answer 1

Here we must know what the difference between the function and the relation from $A{\text{ to }}B$ means. Here we just need to know in brief that if we have inputs and outputs of any numbers or any variables, then if we have only one output for every input then it is called the function and if there are more than one inputs for any variable then it is called the relation from input to output.

Complete step-by-step answer:
Here we are given the three sets which represent as:
$A = \\{ 1,2,3,4\\}$, $B = \\{ 1,5,9,11,15,16\\}$and $f = \\{ (1,5),(2,9),(3,1),(4,5),(2,11)\\}$
Here we need to know that if we have inputs and outputs of any numbers or any variables, then if we have only one output for every input then it is called the function and if there are more than one inputs for any variable then it is called the relation from input to output.
So here we need to keep in mind that if $f$ is the subset of the Cartesian product of the $A \times B$ then we can say that $f$ is the relation from $A{\text{ to }}B$
We know if we have two sets $P = \\{ 1,2,3\\} ,Q = \\{ 4,5\\}$ then $P \times Q = \\{ (1,4),(1,5),(2,4),(2,5),(3,4),(3,5)\\}$
Here we can say that $4$ is the image of $1$ in $(1,4)$ and similarly in all the brackets the second number is the image of the first one. So we need to check it further with relation to $f$
Now we have the set$A = \\{ 1,2,3,4\\}$, $B = \\{ 1,5,9,11,15,16\\}$
So we can say $A \times B = \\{ (1,1),(1,5),(1,9),(1,11),(1,15),(1,16),(2,1),(2,5),(2,9),(2,11),(2,15),(2,16) \\\ {\text{ }}(3,1),(3,5),(3,9),(3,11),(3,15),(3,16),(4,1),(4,5),(4,9),(4,11),(4,15),(4,16)\\} \\\$
Now we are given the function $f = \\{ (1,5),(2,9),(3,1),(4,5),(2,11)\\}$
So first we need to find if $f$ is a relation from $A{\text{ to }}B$
So here we need to keep in mind that if $f$ is the subset of the Cartesian product of the $A \times B$ then we can say that $f$ is the relation from $A{\text{ to }}B$
Now if we compare both $f$ and $A \times B$ we will get that all the elements in the$f$ are the part of $A \times B$
Hence we can say that it is a subset of $A \times B$
So $f$ is a relation from $A{\text{ to }}B$
Now for the function from $A{\text{ to }}B$ one input must have only one output but here in $f$ we can notice that there are two outputs of $2$ which are $\\{ (2,9),(2,11)\\}$
Hence $f$ is not a function from $A{\text{ to }}B$.

Note: Here if we were given the set $f$ as the same set but would have neglected the term $(2,11){\text{ or (2,9)}}$ then it would also be the function from $A{\text{ to }}B$ as here there would be only one image for every number present in the domain.