Chapter 1, Sentential Logic Video Solutions, How to Prove It: A Structured Approach

Problem 1

Analyze the logical forms of the following statements:
(a) We'll have either a reading assignment or homework problems, but we won't have both homework problems and a test.
(b) Either John and Bill are both telling the truth, or neither or them is.
(c) You won't go skiing, or you will and there won't be any snow.
(d) I'll have either fish or chicken, but I won't have both fish and mashed potatoes.
(e) $\sqrt{7} \pm 2$.
(f) 3 is a common divisor of 6,9 , and 15 .

Maria Dearborn

Numerade Educator

01:18

Problem 2

Which of the following expressions are well-formed formulas?
(a) $\neg(\neg P \vee \neg \neg R)$.
(b) $\neg(P, Q, \wedge R)$.
(c) $P \wedge \neg P$.
(d) $(P \wedge Q)(P \vee R)$.

James Kiss

Numerade Educator

02:55

Problem 3

What English sentences are represented by the following expressions?
(a) $\neg(P \wedge \neg S)$, where $P$ stands for the statement "I will buy the pants" and $S$ stands for the statement "I will buy the shirt."
(b) $\neg P \wedge \neg S$, where $P$ and $S$ mean the same thing as in (a).
(c) $(S \vee G) \wedge(\neg S \vee \neg G)$, where $S$ stands for the statement "Steve is happy" and $G$ stands for "George is happy."
(d) $[S \vee(G \wedge \neg S)] \vee \neg G$, where $S$ and $G$ mean the same thing as in (c).

Jennifer Stoner

Numerade Educator

Problem 4

Identify the premises and conclusions of the following deductive arguments and analyze their logical forms. Do you think the reasoning is valid? (Although you will have only your intuition to guide you in answering this last question, in the next section we will develop some techniques for determining the validity of arguments.)
(a) Jane and Pete won't both win the math prize. Pete will win either the math prize or the chemistry prize. Jane will win the math prize. Therefore, Pete will win the chemistry prize.
(b) The main course will be either beef or fish. The vegetable will be either peas or corn. We will not have both fish as a main course and corn as a vegetable. Therefore, we will not have both beef as a main course and peas as a vegetable.
(c) Either John or Bill is telling the truth. Either Sam or Bill is lying. Therefore, either John is telling the truth or Sam is lying.
(d) Either sales will go up and the boss will be happy, or expenses will go up and the boss won't be happy. Therefore, sales and expenses will not both go up.

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