Let M={7,8,9}. Determine which of the following functions is invertible for f:M→M

The correct option is f = {(7,7),(8,8),(9,9)}.

To determine which function is invertible, let us first recall the conditions required for a function to be invertible. A function
$f:M\to M$
is invertible if and only if it is a bijection, which means it must satisfy two conditions:

1. Injective (One-to-One): Each element in the domain maps to a distinct element in the codomain. That is, if
$f\left(x\right)=f\left(y\right)$
then
$x=y$
.

2. Surjective (Onto): Every element in the codomain is mapped to by at least one element in the domain. For a function from a finite set
$M$
to itself, being injective automatically implies being surjective, and vice versa.

Additionally, for a relation to be a valid function, every element in the domain
$M=\{7,8,9\}$
must map to exactly one element in the codomain.

Let us analyze the given options:

Option 1: f = {(7,8),(7,9),(8,9)}
This relation is not a function because the element 7 is mapped to two different values (8 and 9). Furthermore, the domain element 9 has no mapping defined.

Option 2: f = {(9,7),(9,8),(9,9)}
This relation is also not a function because the element 9 is mapped to three different values (7, 8, and 9), while elements 7 and 8 in the domain have no mappings defined.

Option 3: f = {(7,7),(8,8),(9,9)}
In this relation:
- Every element in the domain
$M=\{7,8,9\}$
maps to exactly one element, making it a valid function.
- Each element maps to a unique value (7 maps to 7, 8 maps to 8, 9 maps to 9), meaning the function is injective.
- All elements in the codomain
$M$
are mapped to, meaning the function is surjective.
Since this identity function is bijective, it is invertible.

Option 4: f = {(8,8),(8,7),(9,8)}
This relation is not a function because the element 8 is mapped to two different values (8 and 7), and the domain element 7 has no mapping defined.