How Does MTH500 3.1 Trivial & Vacuous Proofs Work?

Trivial and vacuous proofs are like having a super-easy shortcut to win a game, you don’t even need to play hard.

Imagine you have a toy box with only red balls in it, and someone says: “If there are red balls, then there are balls.” That’s a trivial proof, because we already know all the toys in the box are red balls, so of course there are balls. It’s like saying, “If I eat candy, then I eat something sweet”, it's obvious!

Now imagine you have a toy box with only blue blocks, and someone says: “If there are red balls, then there are blue blocks.” That might sound strange, but it’s actually true because the first part (“if there are red balls”) never happens. We don’t have any red balls, so the condition is like saying, “If pigs fly, then I get ice cream”, since pigs don't fly, you automatically get ice cream! This is a vacuous proof, it's true because the starting part doesn't happen.

So trivial proofs are easy wins, and vacuous proofs are like getting a prize for something that never happens.