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Pre-reading

We will wrap up our discussion of Chapter 1. Read sections 1.6–1.8.

• Section 1.6 (Rules of Inference).The named rules of inference from Table 1 of section 1.6.3 and Table 2 of section 1.6.7 will all be used extensively throughout the course. Make sure you understand how you could verify that each tautology in table 1 is in fact a tautology, and how you can turn each tautology into the corresponding rule of inference.
• Section 1.7 (Introduction to Proofs) and Section 1.8 (Proof Methods and Strategy) are very closely related. We will discuss how we can use the machinery of propositional and predicate logic to form convincing arguments. Most results that we wish to prove are in the form of implications. A theorem typically consists of statements that if some collection of hypotheses are satisfied, then a corresponding conclusion must also hold. Note (as in section 1.7.3) that we often leave out seemingly important words from the statements of theorems, but these words are still understood to be there.
  • We will work through specific examples of Direct proof, Proof by contraposition, Proof by contradiction, Proof by Cases, Existence proofs, and Uniqueness proofs. These techniques will pop up in other mathematics and computer science course throughout your education.
  • Pay attention to the method of "Proof by Cases" (Section 1.8.2). We began investigating the n=2 case of the tautology at the start of that section as Question B2 on assignment 1 and in lecture 4. We will revisit this tautology once we learn about induction.
  • We will look specifically at properties of parity (even and oddness) and how they can be used to establish the irrationality of √2. We will also look at the AM/GM property, and use its extensions to motivate several more advanced techniques later in the course.

A1a (1 point)

(a) Which interpretation of the rule is correct?

• [ ] ∃x(P(x)→Q(x))
• [x] ∀x(P(x)→Q(x))
• [ ] ∀x(P(x)→¬Q(x))
• [ ] ∀x(Q(x)→P(x))
• [ ] none of the above

A1b (2 points)

If we wish to apply the rule to Thomas, who is a 3 years old, which rule (or rules) of inference from sections 1.6.3, 1.6.7, or 1.6.8 lets us conclude that Thomas is not allowed to receive COVID-19 and Flu shots on the same day?