18.17: Logic
Boolean Logic
1. \(\{5,15,25, \ldots\}\)
Quantified Statements
3. At least one person did not fail the quiz today.
Truth Tables
5.
- Elvis is alive or did not gain weight.
- It is not the case that Elvis is alive and gained weight.
- If Elvis gained weight, then he is not alive.
- Elvis is alive if and only if he did not gain weight.
7.
\(\begin{array}{|c|c|c|c|c|}
\hline A & B & \sim A & \sim A \vee B & \sim(\sim A \vee B) \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\
\hline
\end{array}\)
9.
\(\begin{array}{|c|c|c|c|c|c|}
\hline A & B & C & A \vee B & \sim C & (A \vee B) \rightarrow \sim C \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline
\end{array}\)
11.
\(\begin{array}{|c|c|c|}
\hline A & B & A \vee B \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{F} \\
\hline
\end{array}\)
13. The results are identical; the exclusive or translates to " \(\left(A \text { or } B \text { ) and not }(A \text { and } B)^{\prime \prime}\right.\).
Conditional Statements
15.
- Not necessarily true; this is the inverse. You could get your mouth washed out for some other reason.
- True; this is the contrapositive.
- Not necessarily true; this is the converse. You could get your mouth washed out for some other reason.
17. Luke faces Vader and Obi-Wan interferes.
19.
- This couldn’t happen; you fulfilled your part of the bargain but your coach didn’t.
- This couldn’t happen; you didn’t fulfill your part of the bargain but your coach let you play anyway. This could happen with a conditional statement, but not a biconditional.
- This could happen; practice = play, no practice = no play.
De Morgan’s Laws
21. You don’t need a dated receipt or you don’t need your credit card to return this item.
Deductive Arguments
23. Valid, by the law of contraposition.
25. Valid, by disjunctive syllogism.
27. Invalid; we are using the inclusive or, so the sets of people with a pencil and people with a pen could possibly overlap. Marcie might be in the intersection of the two sets.
Logical Fallacies
29. False dilemma; you could fly, take a bus, hitchhike…
31. Correlation implies causation; maybe the only time our smoke detector goes off is when I burn dinner, and the kids choose to eat cereal whenever I burn dinner.