CSCI 301 L12 Worksheet

Lecture 12 - Exercises

Part A - Contrapositive Proof

  1. Prove the following proposition using contrapositive proof.

    For \(x \in \mathbb{R}\), if \(x^2 + 5x < 0\), then \(x < 0\).

  2. Try to prove the proposition from #1 using direct proof; if you reach a point where you’re convinced that this approach is harder, you may give up.

Part B - More Contrapositive Proof

Proposition: Suppose \(a, b \in \mathbb{Z}\). If both \(ab\) and \(a+b\) are even, then both \(a\) and \(b\) are even.

  1. Spend a few moments trying to prove this with direct proof. If you get stuck, stop and move on to #2.
  2. Prove the above proposition using contrapositive proof.

Part C - If you have time

  1. Prove that any odd number is the difference of two squares. (For example, \(7 = 4^2 - 3^2\).)

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.