18.17: Logic

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Boolean Logic

1. \(\{5,15,25, \ldots\}\)

Quantified Statements

3. At least one person did not fail the quiz today.

Truth Tables

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.

\(\begin{array}{|c|c|c|c|c|}
\hline A & B & \sim A & \sim A \vee B & \sim(\sim A \vee B) \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\
\hline
\end{array}\)

\(\begin{array}{|c|c|c|c|c|c|}
\hline A & B & C & A \vee B & \sim C & (A \vee B) \rightarrow \sim C \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline
\end{array}\)

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.