CSCI 301 L12 Notes

Lecture 12 - Notes

Contrapositive Proof

• Know how to use contrapositive proof to show \(P \Rightarrow Q\) by showing that \(\neg Q \Rightarrow \neg P\).
• Practice writing proofs.

Announcements

• A2 due today
• A3 out today (hopefully)
• Resubmission policy amendment: you need to email me and our grader (Ryan, hollobr@wwu.edu) after resubmitting.
  • Reminder of a general communication principle from the syllabus: you cannot assume that I (or the grader or TA) will see or read submission comments; these are not a good way to communicate with us.

The Contrapositive

We have seen: \[ P \Rightarrow Q \equiv \neg Q \Rightarrow \neg P \]

Intuition?

A possible way to build intuition for this is using a statement like:

“All humans are mammals”

which we can rephrase as

“If you are human, then you are a mammal”

If that’s the case, then the following statement rings true to our intuition:

“If you are not a mammal, then you are not human.”

Contrapositive Proof

The fact that \(P \Rightarrow Q\) and its contrapositive are logically equivalent means that showing \(\neg P \Rightarrow \neg Q\) is just as good as showing \(P \Rightarrow Q\). So we can prove \(P \Rightarrow Q\) as follows:

Book of Proof’s Style Guide

1. Begin each sentence with a word, not a mathematical symbol.
2. End each sentence with a period, even when the sentence ends with a mathematical symbol or expression.
3. Separate mathematical symbols and expressions with words.
4. Avoid misuse of symbols.
5. Avoid using unnecessary symbols.
6. Use the first person plural.
7. Use the active voice.
8. Explain each new symbol.
9. Watch out for “it”, unless it’s completely unambiguous what “it” refers to.
10. Since, because, as, for, so are used to mean that P is true (or assumed to be true) and as a consequence Q is true also.
11. Thus, hence, therefore, consequently precede a statement that follows logically from previous sentences or clauses.
12. Clarity is the gold standard of mathematical writing.

Definitions

Definition: An integer \(n\) is even if \(n = 2a\) for some integer \(a \in \mathbb{Z}\).

Observation: by (confusing) convention, definitions are usually stated as a conditional but meant to imply a biconditional. That is, we interpret the above definition to mean: \[ n \text{ is even} \Leftrightarrow \exists a \in Z, n = 2a \] What is the distinction? We are saying that \(n\) is not even if it does not satisfy the given property, where the unidirectional implication would be inconclusive about that case.

Some more definitions, just so we have something to play with in proofs.

Definition: An integer \(n\) is odd if \(n = 2a + 1\) for some integer \(a \in \mathbb{Z}\).

Definition: Suppose \(a\) and \(b\) are integers. We say that \(a\) divides \(b\), written \(a | b\), if \(b = ac\) for some \(c \in \mathbb{Z}\). In this case we also say that \(a\) is a divisor of \(b\), and that \(b\) is a multiple of \(a\).

Definition: Two integers have the same parity if they are both even or they are both odd. Otherwise they have opposite parity.