More Logic: Equivalence, Conditionals, and Biconditionals

As we enter the second week, we are going to pick up the pace on introducing videos, readings, and warmup problems.

Videos (~35 mins)

Reading (40 mins)

Optional Reading

Warmup Problems (~40 mins)

Please complete these problems “by hand,” without typing. Pencil/pen and paper is best practice for the quiz, but using a stylus with a tablet is also fine. Then, scan/photograph or otherwise produce an image of your solutions and upload them to Canvas.

Problem 1

Consider the following propositions:

  • \(p\): I achieved a Satisfactory (S) assessment on at least 18 Learning Targets.
  • \(q\): I received an E on at least 4 Lab assignments.
  • \(r\): I had an N or an R on at least one Lab assignment by the end of the semester.
  • \(s\): I used 4 or more participation passes.
  • \(t\): I earned an A in CSCI 0200.

Construct the most precise logical statement that you can, using \(\lnot\), \(\lor\), \(\land\), \(\rightarrow\), and/or \(\leftrightarrow\), to describe the relationship between these five propositions.

Note: there are several correct logical statements you could construct, but there is one that is the most precise, specific, and informative about how this class works.

Problem 2

Consider the expression \(\lnot p \land (q \rightarrow p)\).

Use logical equivalences to simplify this expression as far as you can. Please write your chain of logical equivalences in standard form. This means that you should cite a previously-known logical equivalence with each of your steps, on the righthand side of each line.

If you knew this expression was true, what could you conclude about \(q\)?

Hint: Get everything in terms of \(\land\), \(\lor\), and \(\lnot\) and then use the following two rules of logical equivalence:

  • Distributive Law: \(a \land (b \lor c) \equiv (a \land b) \lor (a \land c)\).
  • Law of Excluded Middle: \(a \land \lnot a \equiv \mathrm{F}\), i.e. \(a \land \lnot a\) is a logical contradiction.

Problem 3

We often write the biconditional “if and only if” as \(p \leftrightarrow q\), suggesting an implication that goes both ways. Make a truth table for \(p \leftrightarrow q\) by making a detailed truth table for its expanded form, \((p \rightarrow q) \land (q \rightarrow p)\).



© Phil Chodrow, 2024