Proofs of existential statements often boil down to finding an example that makes the predicate true. Sometimes these proofs look a bit silly as formal proofs, but they can still be written in that form. For example….
Proof. We will show that there exists a real number \(x\) such that \(3x^2 + 1 = 13\) by solving for such a value:
We have thus found that \(2\) and \(-2\) are real numbers that satisfy \(3x^2 + 1 = 13\). Therefore we see that there exists a real number \(x\) such that \(3x^2 + 1 = 13\). □
Proofs about universal quantifiers, on the other hand, tend to look more like what we’re used to. For example….
Theorem 2. Every even integer can be written in the form \(n + 2\) where \(n\) is another even integer.
Proof. We assume that \(m\) is an even integer, and will show that \(m = n+2\) for some integer \(n\), thereby establishing that every even integer can be written in the form \(n + 2\) where \(n\) is another even integer. From the definition of even integer, \(m = 2b\) for some integer \(b\). Letting \(n = m - 2\), we see that \(m = n + 2\), and so we only need to show that \(n\) is also even. Substituting \(m = 2b\) into the definition of \(n\) gives
where \(a = b - 1\). Because the integers are closed under subtraction and \(b\) is an integer, so is \(a\), and so \(n\) is even. We have now shown that \(m\) can be written in the form \(n + 2\) for some even integer \(n\), and thus that every even integer is of the form \(n + 2\) where \(n\) is another even integer. □