CSCI 301 L08 Notes

Lecture 8 - 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.

Modular Equivalence

• Know and be able to apply the definition of congruence modulo \(\mathbf{n}\)

Announcements

• Event of interest:

  

  • Separate Q&A at 5:45 (after the main event) for students interested in Data Science careers.
    • Pizza provided - RSVP here so they order enough.

• Midterm exam date set: Monday, 5/5 in class.

• A2 due Friday

• Week 2 Feedback themes:

  • Lab is challenging - yep!
  • Group work, exercises, notes, etc. are helpful - yay!

Modular Congruence

Definition: Given integers \(a\) and \(b\) and \(n \in \mathbb{Z}\), we say that \(a\) and \(b\) are congruent modulo \(\mathbf{n}\) if \(n \mid (a - b)\). We write this as \(a \equiv b \pmod{n}\). If \(a\) and \(b\) are not congruent modulo \(n\), then we write \(a \not\equiv b \pmod{n}\).

Do Exercises Part A

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:

Example

Proposition: Suppose \(x, y \in \mathbb{Z}\). If \(5 \nmid xy\), then \(5 \nmid x\) and \(5 \nmid y\).

See whiteboard notes for proof.

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 \(a \in \mathbb{Z}\).

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.

Definition: Given integers \(a\) and \(b\) and \(n \in \mathbb{Z}\), we say that \(a\) and \(b\) are congruent modulo \(\mathbf{n}\) if \(n \mid (a - b)\). We write this as \(a \equiv b \pmod{n}\). If \(a\) and \(b\) are not congruent modulo \(n\), then we write \(a \not\equiv b \pmod{n}\).