Chapter 1 - Section 1.3 - Propositional Equivalences - Exercises - Page 35: 9

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a) The conditional statement is a tautology because the last column of the following truth table contains only True values. $\underline{p\quad q \quad p \land q \quad (p\land q)\rightarrow p }$ $T\quad T \quad T \quad\quad\quad T $ $T\quad F \quad F \quad\quad\quad T $ $F\quad T \quad F \quad\quad\quad T $ $T\quad F \quad F \quad\quad\quad T $ b) The conditional statement is a tautology because the last column of the following truth table contains only True values. $\underline{p\quad q \quad p \lor q \quad p\rightarrow (p \lor q) }$ $T\quad T \quad T \quad\quad\quad T $ $T\quad F \quad T \quad\quad\quad T $ $F\quad T \quad T \quad\quad\quad T $ $T\quad F \quad F \quad\quad\quad T $ c) The conditional statement is a tautology because the last column of the following truth table contains only True values. $\underline{p\quad q \quad \neg p \quad p \rightarrow q \quad \neg p\rightarrow (p \rightarrow q) }$ $T\quad T \quad F \quad \quad T \quad \quad \quad \quad T $ $T\quad F \quad F \quad \quad F \quad \quad \quad \quad T $ $F\quad T \quad T \quad \quad T \quad \quad \quad \quad T $ $F\quad F \quad T \quad \quad T \quad \quad \quad \quad T $ d) The conditional statement is a tautology because the last column of the following truth table contains only True values. $\underline{p\quad q \quad p\land q \quad p \rightarrow q \quad (p \land q)\rightarrow (p \rightarrow q) }$ $T\quad T \quad T \quad \quad T \quad\quad \quad \quad \quad T $ $T\quad F \quad F \quad \quad F \quad\quad \quad \quad \quad T $ $F\quad T \quad F \quad \quad T \quad\quad \quad \quad \quad T $ $F\quad F \quad F \quad \quad T \quad\quad \quad \quad \quad T $ e) The conditional statement is a tautology because the last column of the following truth table contains only True values. $\underline{p\quad q \quad p\rightarrow q \quad \neg (p \rightarrow q) \quad \neg (p \rightarrow q)\rightarrow p }$ $T\quad T \quad T \quad \quad \quad \quad F \quad \quad \quad \quad\quad \quad T $ $T\quad F \quad F \quad \quad \quad \quad T \quad \quad \quad \quad\quad \quad T $ $F\quad T \quad T \quad \quad \quad \quad F \quad \quad \quad \quad\quad \quad T $ $F\quad F \quad T \quad \quad \quad \quad F \quad \quad \quad \quad\quad \quad T $ f) The conditional statement is a tautology because the last column of the following truth table contains only True values. $\underline{p\quad q \quad p\rightarrow q \quad \neg (p \rightarrow q) \quad \neg q \quad \neg (p \rightarrow q)\rightarrow \neg q }$ $T\quad T \quad T\quad\quad\quad\quad F\quad\quad \quad F \quad\quad\quad T$ $T\quad F \quad F\quad\quad\quad\quad T\quad\quad \quad T \quad\quad\quad T$ $F\quad T \quad T\quad\quad\quad\quad F\quad\quad \quad F \quad\quad\quad T$ $F\quad F \quad T\quad\quad\quad\quad F\quad\quad \quad T \quad\quad\quad T$