1. ## Function

List all functions from $\displaystyle \{a,b\}$ to $\displaystyle \{1,2\}$.

For some reason I keep missing one!! I know that I should have $\displaystyle 2^2$ functions.

So I start by mapping a to 1 and 2, leaving b alone.

Then I map b to 1 and 2, leaving a alone.

Then I map a to 1, and b to 2.

But i'm missing one!

Is it a null function? One that doesn't map a and b to anything?

2. $\displaystyle \left\{ {(a,1),(b,1)} \right\},~\left\{ {(a,1),(b,2)} \right\},~\left\{ {(a,2),(b,1)} \right\},~\&~\left\{ {(a,2),(b,2)} \right\}$

3. Originally Posted by Showcase_22
List all functions from $\displaystyle \{a,b\}$ to $\displaystyle \{1,2\}$.

For some reason I keep missing one!! I know that I should have $\displaystyle 2^2$ functions.

So I start by mapping a to 1 and 2, leaving b alone.

Then I map b to 1 and 2, leaving a alone.
Neither of those are functions!

Then I map a to 1, and b to 2.

But i'm missing one!

Is it a null function? One that doesn't map a and b to anything?
I think you need to review the definition of "function".

$\displaystyle 1\to a, 2\to a$
$\displaystyle 1\to b, 2\to b$
$\displaystyle 1\to a, 2\to b$
$\displaystyle 1\to b, 2\to a$

Plato did it to me again- got his response in while I was still typing mine!