Chapter 1

Mark each statement True or False. Justify each answer. (a) In order to be classified as a statement, a sentence must be true. (b) Some statements are both true and false. (c) When statement \(p\) is true, its negation \(\sim p\) is false. (d) A statement and its negation may both be false. (e) In mathematical logic, the word "or" has an inclusive meaning.

Mark each statement True or False. Justify each answer. (a) In an implication \(p \Rightarrow q\), statement \(p\) is referred to as the proposition. (b) The only case where \(p \Rightarrow q\) is false is when \(p\) is true and \(q\) is false. (c) "If \(p\), then \(q\) " is equivalent to " \(p\) whenever \(q\) ". (d) The negation of a conjunction is the disjunction of the negations of the individual parts. (e) The negation of \(p \Rightarrow q\) is \(q \Rightarrow p\).

Write the negation of each statement. it (a) The \(3 \times 3\) identity matrix is singular. (b) The function \(f(x)=\sin x\) is bounded on \(\mathbb{R}\). (c) The functions \(f\) and \(g\) are linear. (d) Six is prime or seven is odd. (e) If \(x\) is in \(D\), then \(f(x)<5\). (f) If \(\left(a_{n}\right)\) is monotone and bounded, then \(\left(a_{n}\right)\) is convergent. (g) If \(f\) is injective, then \(S\) is finite or denumerable.

Write the negation of each statement. (a) The function \(f(x)=x^{2}-9\) is continuous at \(x=3\). (b) The relation \(R\) is reflexive or symmetric. (c) Four and nine are relatively prime. (d) \(x\) is in \(A\) or \(x\) is not in \(B\). (e) If \(x<7\), then \(f(x)\) is not in \(C\). (f) If \(\left(a_{n}\right)\) is convergent, then \(\left(a_{n}\right)\) is monotone and bounded. (g) If \(f\) is continuous and \(A\) is open, then \(f^{-1}(A)\) is open.

Identify the antecedent and the consequent in each statement. t (a) \(M\) has a zero eigenvalue whenever \(M\) is singular. (b) Linearity is a sufficient condition for continuity. (c) A sequence is Cauchy only if it is bounded. (d) \(x<3\) provided that \(y>5\).

Identify the antecedent and the consequent in each statement. (a) A sequence is convergent if it is Cauchy. (b) Convergence is a necessary condition for boundedness. (c) Orthogonality implies invertability. (d) \(K\) is closed and bounded only if \(K\) is compact.

Identify the antecedent and the consequent in each statement. (a) A sequence is convergent if it is Cauchy. (b) Convergence is a necessary condition for boundedness. (c) Orthogonality implies invertability. (d) \(K\) is closed and bounded only if \(K\) is compact.

Construct a truth table for each statement. (a) \(p \vee \sim q\) (b) \(p \wedge \sim p\) (c) \([(\sim q) \wedge(p \Rightarrow q)] \Rightarrow \sim p\)

Indicate whether each statement is True or False. t? (a) \(3 \leq 5\) and 11 is odd. (b) \(3^{2}=8\) or \(2+3=5\). (c) \(5>8\) or 3 is even. (d) If 6 is even, then 9 is odd. (e) If \(8<3\), then \(2^{2}=5\). (f) If 7 is odd, then 10 is prime. (g) If 8 is even and 5 is not prime, then \(4<7\). (h) If 3 is odd or \(4>6\), then \(9 \leq 5\). (i) If both \(5-3=2\) and \(5+3=2\), then \(9=4\). (j) It is not the case that 5 is even or 7 is prime.

Indicate whether each statement is True or False. (a) \(2+3=5\) and 5 is even. (b) \(3+4=5\) or \(4+5=6\). (c) 7 is even or 6 is not prime. (d) If \(4+4=8\), then 9 is prime. (e) If 6 is prime, then \(8<6\). (f) If \(6<2\), then \(4+4=8\). (g) If 8 is prime or 7 is odd, then 9 is even. (h) If \(2+5=7\) only if \(3+4=8\), then \(3^{2}=9\). (i) If both \(5-3=2\) and \(5+3=8\), then \(8-3=4\). (j) It is not the case that 5 is not prime and 3 is odd.

Let \(p\) be the statement "The figure is a polygon," and let \(q\) be the statement "The figure is a circle." Express each of the following statements in symbols. t' (a) The figure is a polygon, but it is not a circle. (b) The figure is a polygon or a circle, but not both. (c) If the figure is not a circle, then it is a polygon. (d) The figure is a circle whenever it is not a polygon. (e) The figure is a polygon iff it is not a circle.

Define a new sentential connective \(\nabla\), called nor, by the following truth table. \begin{tabular}{c|c|c} \hline\(p\) & \(q\) & \(p \nabla q\) \\ \hline \(\mathrm{T}\) & \(\mathrm{T}\) & \(\mathrm{F}\) \\ \(\mathrm{T}\) & \(\mathrm{F}\) & \(\mathrm{F}\) \\ \(\mathrm{F}\) & \(\mathrm{T}\) & \(\mathrm{F}\) \\ \(\mathrm{F}\) & \(\mathrm{F}\) & \(\mathrm{T}\) \\ \hline \end{tabular} (a) Use a truth table to show that \(p \nabla p\) is logically equivalent to \(\sim p\). (b) Complete a truth table for \((p \nabla p) \nabla(q \nabla q)\). (c) Which of our basic connectives \((p \wedge q, p \vee q, p \Rightarrow q, p \Leftrightarrow q)\) is logically equivalent to \((p \nabla p) \nabla(q \nabla q)\) ?

Use truth tables to verify that each of the following is a tautology. Parts (a) and (b) are called commutative laws, parts (c) and (d) are associative laws, and parts (e) and (f) are distributive laws. (a) \((p \wedge q) \Leftrightarrow(q \wedge p)\) (b) \((p \vee q) \Leftrightarrow(q \vee p)\) (c) \([p \wedge(q \wedge r)] \Leftrightarrow[(p \wedge q) \wedge r]\) (d) \([p \vee(q \vee r)] \Leftrightarrow[(p \vee q) \vee r]\) (e) \([p \wedge(q \vee r)] \Leftrightarrow[(p \wedge q) \vee(p \wedge r)]\) (f) \([p \vee(q \wedge r)] \Leftrightarrow[(p \vee q) \wedge(p \vee r)]\)