CSCI 301 L06 Notes

Lecture 6 - Notes

Logic 2

• articulate the meanings of logical AND, OR, NOT, conditional statements and biconditional statements.
• use truth tables to calculate truth values of logical statements
• use truth tables to verify logically equivalent statements

Announcements

• Reminder: fill out Week 1 survey by tonight

• A2 - details TBA later today.

Statements, Revisited

How can a sentence be a statement (i.e., definitely true or definitely false) without us knowing which?

Example: The Collatz Conjecture

Start with a positive integer \(x\). Repeat the following:

• if \(x\) is even, divide it by 2 (\(x \leftarrow x / 2\))
• if \(x\) is odd, multiply \(x\) by 3 and add 1 (\(x \leftarrow 3x + 1\))

Given any initial positive integer \(x\), after iterating this a finite number of times, \(x\) becomes 1.

This is a statement (it’s either true or not). But nobody has figured out which!

Logic, Continued

See also whiteboard notes

Conditional (if/then) statements

• Truth table logic puzzle pt 4

Example: If \(x = y\), then \(x^2 = y^2\).

Notice: this does not commute.

• \(P\) : \(x = y\)
• \(Q\) : \(x^2 = y^2\)
• \(R\) : If \(P\) then \(Q\) (a true statement)
• \(S\) : If \(Q\) then \(P\) (a false statement)

False Implies Anything Weirdness

Notice: If false, then anything! This gets weird sometimes.

\(P\) : If I am healthy, then I will come to class.

Suppose you are sick; then the statement “If I am healthy, then I will come to class” is true whether or not you come to class.

Mental model suggestions:

• Apply the truth table and don’t think too hard
• Think of \(P \Rightarrow Q\) as a promise that anytime \(P\) is true, \(Q\) will also be true; but it doesn’t promise any more than that.

Do Exercises Part B

Natural Language Weirdness

Equivalent ways of writing “if \(P\) then \(Q\)”:

• If \(P\), then \(Q\)
• \(Q\) if \(P\)
• \(Q\) whenever \(P\)
• Whenever \(P\), then also \(Q\).
• \(P\) is a sufficient condition for \(Q\).
• \(Q\) is a necessary condition for \(P\).
• \(P\) only if \(Q\).

Definition/aside: “If \(Q\) then \(P\)” is the converse of “If \(P\) then \(Q\)”.

Do Exercises Part C

Biconditional - “if and only if”: \(P \Leftrightarrow Q\) means \(P \Rightarrow Q\) and \(Q \Rightarrow P\) are both true.

Do exercises Part D