The 2 foundational ways of counting/Statements

Foundations

Let us be given two decisions:

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

We require that:

• There always be [ilmath]m[/ilmath] options for [ilmath]A[/ilmath] and [ilmath]n[/ilmath] options for [ilmath]B[/ilmath] 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 [ilmath]m[/ilmath] for [ilmath]A[/ilmath] and [ilmath]n[/ilmath] for [ilmath]B[/ilmath]

Then:

And

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

• [ilmath]\mathcal{O}:\eq mn[/ilmath]

Xor

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

• [ilmath]\mathcal{O}:\eq m+n[/ilmath]