Mathematical rigour in Combinatorics

Question

I understand these statements but they lack rigour. How do you define these statements rigorously?

1. If there are $a$ varieties of soup and $b$ varieties of salad, then there are $a + b$ possible ways to order a meal of soup or salad.

2. If there are $a$ varieties of soup and $b$ varieties of salad then there are $ab$ possible ways to order a meal of soup and salad.

Your help will be appreciated rather than taking the time to downvote the question.

Answer 1

Let $A$ the set of soup varieties, and $B$ be the set of salad varieties, where $|A|=a$ and $|B|=b$ (where $A\cap B=\varnothing$). Then $|A\cup B|=a+b$ and $|A\times B|=ab$.

Answer 2

I don't see how these statements aren't rigorous enough. Here are some obtuse ways to make them more so:

1. Let $A$ be the set of varieties of soup ($|A|=a$) and $B$ the set of varieties of salad ($|B|=b$). Then, since $A\cap B=\emptyset$, $|A\cup B|=a+b$.

2. Let $A$ be the set of varieties of soup ($|A|=a$) and $B$ the set of varieties of salad ($|B|=b$). Then, $|A\times B|=ab$.