I’m asked to prove the statement in the title under the assumption that I do not know the Inclusion-Exclusion Principle. I have two ways of starting the proof where:

- I could declare two sets with a certain amount of values and show by example that it is true:

A = {1. 2, 3, 4} and B = {3, 4, 5, 6}

|A| = 4, |B| = 4

|A ∪ B| = |A| + |B| – |A ∩ B| = 4 + 4 – 2 = 6

- I could state that it is true and give a logical explanation:

This is true, because to count the number of elements in A ∪ B, we start by counting those in A, and then add those in B. If A and B were disjoint, then we are done, otherwise, we have double counted those in both sets, so we must subtract those in A ∩ B.

However, I don’t know if these are counted as formal proofs. If not, how would I start a proof like this?