Chapter 3 Proof Techniques
Many mathematical theorems are universally quantified. For example, in linear algebra, we can prove the theorem, “For all \(n \times n\) matrices \(A\text{,}\) \(A\) is nonsingular if and only if \(A^\top\) is nonsingular.” This chapter introduces three elementary proof techniques.
To prove that something is true for all \(x \in D\text{,}\) we begin with “Let \(x \in D\)”--that is, we consider a generic element of \(D\text{,}\) not a a specific element of \(D\) (i.e., not an example). After invoking a generic element, there are several techniques for proving such theorems:
- Direct Proof: working with the statement as-is
- Indirect Proof: modifying the statement before working with it (proof by contrapositve, proof by contradiction, proof by other logical equivalences)
