The 2 foundational ways of counting/Statements

Foundations

Let us be given two decisions:

• The first, $A$, for which we have $m\in\mathbb{N}_{\ge 0}$ outcomes
• The second, $B$, for which we have $n\in\mathbb{N}_{\ge 0}$ outcomes.

We require that:

• There always be $m$ options for $A$ and $n$ options for $B$ irrespective of which we decide first, and irrespective of what we decide first.
  • This does not mean the options cannot be different depending on which we tackle first and what we select, just that there must always be $m$ for $A$ and $n$ for $B$

Then:

And

If we must choose once from $A$ and once from $B$, then the number of outcomes, $\mathcal{O}\in\mathbb{N}_{\ge 0}$ is:

• $\mathcal{O}:\eq mn$

Xor

If we must choose from either $A$ or $B$ - but not both, then the number of outcomes, $\mathcal{O}\in\mathbb{N}_{\ge 0}$ is:

• $\mathcal{O}:\eq m+n$