Let X = {1, 2, 3, 4,}, Y = {1, 5, 9, 11, 15, 16}

Question:

Let X = {1, 2, 3, 4,}, Y = {1, 5, 9, 11, 15, 16} and F = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}.

Are the following true?

(i) F is a relation from X to Y (ii) F is a function from X to Y. Justify your answer in following true?

Solution:

X = {1, 2, 3, 4} and Y = {1, 5, 9, 11, 15, 16}

and F = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}

(i) To show: $F$ is a relation from $X$ to $Y$

First elements in $\mathrm{F}=1,2,3,4$

All the first elements are in Set $X$

So, the first element is from set $X$

Second elements in $\mathrm{F}=5,9,1,11$

All the second elements are in Set $Y$

So, the second element is from set $Y$

Since the first element is from set $X$ and the second element is from set $Y$

Hence, $F$ is a relation from $X$ to $Y$.

(ii) To show: $F$ is a function from $X$ to $Y$

Function:

(i) all elements of the first set are associated with the elements of the second set.

(ii) An element of the first set has a unique image in the second set.

$F=\{(1,5),(2,9),(3,1),(4,5),(2,11)\}$

Here, 2 is coming twice.

Hence, it does not have a unique (one) image.

So, it is not a function.