CSCI 301 L06 Notes

Lecture 6 - Notes

Logic 3

• know (or know of the existence of so you can later look up) some useful equivalences, including the contrapositive, de Morgan’s laws, and negation of quantifiers and conditionals
• Be able to negate statements expressed symbolically.

Announcements

• Reminder: fill out Week 2 survey by tonight
• CS Tutors: CF 162/164, M-F 4-7pm

Equivalences

You can check whether two statements are equivalent by seeing if their truth tables match exactly.

Example: Does \(\land\) distribute over \(\lor\)? In other words

\(A \land (B \lor C) \stackrel{?}{\equiv} (A \land B) \lor (A \land C)\)

Do Exercises Part A

Negation

In the last exercises, we saw some important equivalences that allow us to negate common expressions.

#4-5 - De Morgan’s laws can negate statements involving the and \((\land)\) and or \((\lor)\) operators: \[ \begin{align*} \neg{(P \land Q) \equiv (\neg P) \lor (\neg Q)}\\ \neg{(P \lor Q) \equiv (\neg P) \land (\neg Q)} \end{align*} \]

#6 - negate a conditional statement: \[ \neg (P \Rightarrow Q) \equiv P \land \neg Q \]

What about negating quantifiers - how would you negate each of the following?

• \(\exists x, P(x)\)
• \(\forall x, P(x)\)

See also the equivalences reference sheet.

Do Exercises Part B

Logical Inference

Given the truth value of one or more, deduce the truth value of another statement. Most of these are intuitive and you don’t need to memorize the names, but here they are:

• Modus Ponens: if you know \(P \Rightarrow Q\) and \(P\) is true, you can infer that \(Q\) is true.
• Modus Tollens: if you know \(P \Rightarrow Q\) and \(Q\) is not true, you can infer that \(P\) is not true.
• Elimination: if you know that \(P \lor Q\) and \(P\) is not true, you can infer that \(Q\) is true.