Implication, Inverse, Converse, and Contrapositive

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

Videos (~35 mins)

Tautologies and Contradictions (3:40)
3 Ways to Show a Logical Equivalence (5:28)
Conditional Statements: if p then q (7:09)
Vacuously True Statements (2:00)
Negating a Conditional Statement (2:51)
Contrapositive of a Conditional Statement (4:59)
The Converse and Inverse of a Conditional Statement (5:04)
Biconditional Statements | “if and only if” (2:53)

Reading (40 mins)

DMOI 3.1 through Example 3.1.5.

Optional Reading

BOP 2.3-2.6

Warmup Problems (~30 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 11 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 missed 4 or more warmups.
\(T\): I earned an A in CSCI 0200.

Construct a correct logical statement using \(\lnot\), \(\lor\), \(\land\), \(\rightarrow\), and/or \(\leftrightarrow\) to describe the relationship between these five propositions.

Problem 2

Consider the expression \(\lnot P \land (Q \rightarrow P)\).

Use logical equivalences to simplify this expression as far as you can. 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 \(p \leftrightarrow q\). Make a truth table for \(p \leftrightarrow q\) by making a detailed truth table for its expanded form, \((p \rightarrow q) \land (q \rightarrow p)\).