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Chapter 1, #3 (h) : Translate the following sentence into symbolic notation:
"If John testifies and tells the truth, he will be found guilty; and if he does not testify, he will be found guilty."
Solution: Let $P$ be "John testifies.", $Q$ be "John tells the truth", $R$ be "John will be found guilty.", and $S$ be "John does not testify." Then this sentence translates to
$$[(P \wedge Q) \rightarrow R] \wedge (\neg P \rightarrow R).$$
Chapter 1, #4 (a): Determine the truth value of the sentence
"If Galileo was born before Descartes, then Newton was not born before Shakespeare."
Solution: Let $P = ``$Galileo was born before Descartes$"$ and $Q=``$ Newton was not born before Shakespeare$"$. As we are told in this problem, $P$ is true and "Newton was born before Shakespeare" is false (so $Q$, which is the negation of this statement, is true). The given sentence, written symbolicly, is
$$P \rightarrow Q.$$
If we substituting in the truth values that we are given, this yields $T \rightarrow T$, which is true (by the implication truth table).
Chapter 1, #5 (d): If $N$ and $W$ are true and $C$ is false, is the following sentence true or false?
$$N \leftrightarrow (\neg W \vee C)$$
Solution: Substitute the truth values in and calculate using truth tables:
$$T \leftrightarrow (\neg T \vee F)$$
$$T \leftrightarrow (F \vee F)$$
$$T \leftrightarrow F,$$
which is false.
Chapter 1, #6 (e): If $P, Q,$ and $R$ are true and $S$ is false, is the following sentence true or false?
$$(P \wedge Q) \wedge (R \wedge S)$$
Solution: Substitute in the truth values and calculate using truth tables:
$$(T \wedge T) \wedge (T \wedge F)$$
$$T \wedge F,$$
which is false.
Chapter 1, #6 (i): If $P, Q,$ and $R$ are true and $S$ is false, is the following sentence true or false?
$$(P \rightarrow \neg Q) \rightarrow (S \leftrightarrow R)$$
Solution: Substitute in the truth values to get
$$(T \rightarrow \neg T) \rightarrow (F \leftrightarrow T)$$
$$(T \rightarrow F) \rightarrow F$$
$$F \rightarrow F,$$
which is true.
