Let p be an odd prime. Show that the congruence $x^{4} \equiv-1(\bmod p)$ has a solution if and only if p is of the form 8k + 1.

Answer “true” or “false” to the following statements: Some arguments, while not completely valid, are almost valid.

Construct a truth table for each of the following compound statements, where p, q, r denote primitive statements. a) $\neg(p \vee \neg q) \rightarrow \neg p$ b) $p \rightarrow(q \rightarrow r)$ c) $(p \rightarrow q) \rightarrow r$ d) $(p \rightarrow q) \rightarrow(q \rightarrow p)$ e) $[p \wedge(p \rightarrow q)] \rightarrow q$ f) $(p \wedge q) \rightarrow p$ g) $q \leftrightarrow(\neg p \vee \neg q)$ h) $[(p \rightarrow q) \wedge(q \rightarrow r)] \rightarrow(p \rightarrow r)$

Show that each of these conditional statements is a tautology by using truth tables. a) [¬p ∧ (p ∨ q)] → q b) [(p → q) ∧ (q → r)] → (p → r) c) [p ∧ (p → q)] → q d) [(p ∨ q) ∧ (p → r) ∧ (q → r)] → r

Determine whether each of these compound propositions is satisfiable.

a) (p ∨ q ∨ ¬r) ∧ (p ∨ ¬q ∨ ¬s) ∧ (p ∨ ¬r ∨ ¬s) ∧ (¬p ∨ ¬q ∨ ¬s) ∧ (p ∨ q ∨ ¬s)

b) (¬p ∨ ¬q ∨ r) ∧ (¬p ∨ q ∨ ¬s) ∧ (p ∨ ¬q ∨ ¬s) ∧ (¬p ∨ ¬r ∨ ¬s) ∧ (p ∨ q ∨ ¬r) ∧ (p ∨ ¬r ∨ ¬s)

c) (p ∨ q ∨ r) ∧ (p ∨ ¬q ∨ ¬s) ∧ (q ∨ ¬r ∨ s) ∧ (¬p ∨ r ∨ s) ∧ (¬p ∨ q ∨ ¬s) ∧ (p ∨ ¬q ∨ ¬r) ∧ (¬p ∨ ¬q

∨ s) ∧ (¬p ∨ ¬r ∨ ¬s)

Prove that $\sqrt[3]{2}$ is irrational.