Question

Prove that if 6 people find themselves in an elevator together, then it must be true that at least 3 of those people know each other, or at least 3 of those people don't know each other 6.

Answer 1

This proof comes by complement;
There are 6 people in the elevator;

Let us assume that : At least 3 of those people know each other is FALSE
In that case, no 3 people know each other. So out of 6 people, there will be at least 3 people who do not know each other since you can choose any 3 out of those 6 people and (as per assumption) there is no way that those 3 people selected will know each other. This implies that there are at least 3 people who do not know each other.

Now reject the assumption and we get that at least 3 of those people know each othe is true. In which case, it still satisfies the first part of the OR statement in the question.