Chapter 1 - Section 1.3 - Propositional Equivalences - Exercises - Page 36: 50

See step by step answer for solution.

a) Two propositions are logically equivalent if they have the same truth value for any combination of truth values of the variables. A conjunction $p \downarrow q$ is true if and only if both p and q are false. A conjunction $\neg p $ is true if p is false. $\underline{p \quad p\downarrow p \quad \neg p}$ $T \quad F\quad\quad\quad F$ $F \quad T\quad\quad\quad T$ Since the last two columns of the truth table contain the same truth value in every row, the last two expressions are logically equivalent. b) $\underline{p \quad q \quad p\downarrow q \quad (p\downarrow q)\downarrow (p\downarrow q) \quad p \lor q}$ $T \quad T \quad F \quad\quad\quad\quad\quad T\quad\quad\quad\quad \quad T$ $T \quad F \quad F \quad\quad\quad\quad\quad T\quad\quad\quad\quad \quad T$ $F \quad T \quad F \quad\quad\quad\quad\quad T\quad\quad\quad\quad \quad T$ $F \quad F \quad T \quad\quad\quad\quad\quad F\quad\quad\quad\quad \quad F$ Since the last two columns of the truth table contain the same truth value in every row, the last two expressions are logically equivalent. c) The set {$ \neg, \lor $} if functionally complete and we have proved that both $\neg$ and $ \lor$ can be represented with $\downarrow$ ($ \neg p \equiv p \downarrow p$ and $p \lor q \equiv (p\downarrow q)\downarrow (p\downarrow q)$). So, we conclude that {$\downarrow$} is a functionally complete collection.