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- Question : 1E - Use your knowledge of number systems to determine whether each is true or false. (a) 11 is a rational number. ? (b) 5? is a rational number. (c) There are exactly 3 prime numbers between 40 and 50. (d) There are exactly 5 prime numbers less than 10. (e) 29 is a composite number. (f) 0 is a natural number. ? (g) (5 + 2i )(5 ? 2i) is a real number. (h) 18 is a multiple of 12.
- Question : 2E - Which of the following are propositions? Give the truth value of each proposition. (a) What time is dinner? (b) It is not the case that 5 + ? is not a rational number. ? (c) x/2 is a rational number. (d) 2x + 3y is a real number. (e) Either 3 + ? is rational or 3 ? ? is rational. ? (f) Either 2 is rational and ? is irrational, or 2? is rational. (g) Either 5? is rational and 4.9 is rational, or 3? is rational. (h) ? 12 is rational, and either 3? < 10 or 3? > 15. (i) It is not the case that 39 is prime, or that 64 is a power of 2. (j) There are more than three false statements in this book and this state- ment is one of them.
- Question : 3E - Make truth tables for each of the following propositional forms. ? (a) P ? ?P. (b) P ? ?P. ? (c) P ? ?Q. (d) P ? (Q ? ?Q). ? (e) (P ? Q) ? ?Q. (f) ?(P ? Q). (g) (P ? ?Q ) ? R. (h) ?P ? ?Q. ? (i) P ? (Q ? R). (j) (P ? Q) ? (P ? R). (k) P ? P. (l) (P ? Q) ? (R ? ?S).
- Question : 4E - If P, Q, and R are true while S and T are false, which of the following are true? ? (a) Q ? (R ? S). (b) Q ? (R ? S). ? (c) (P ? Q ) ? (R ? S). (d) (?P ? ?Q) ? (?R ? ?S). (e) ?P ? ?Q. ? (f) (?Q ? S) ? (Q ? S). ? (g) (P ? S) ? (P ? T).
- Question : 5E - Use truth tables to prove the remaining parts of Theorem 1.1.1.
- Question : 6E - Which of the following pairs of propositional forms are equivalent? ? (a) P ? P, P. (b) P ? P, P. ? (c) P ? Q, Q ? P. (d) (?P) ? (?Q), ? (P ? ?Q). ? (e) ?P ? ?Q, ?(P ? ?Q). (f) ?(P ? Q), ?P ? ?Q. ? (g) (P ? Q ) ? R, P ? (Q ? R). (h) (P ? Q) ? R, P ? (Q ? R).
- Question : 7E - Determine the propositional form and truth value for each of the following: (a) It is not the case that 2 is odd. (b) f (x) = e x is increasing and concave up. (c) Both 7 and 5 are factors of 70. (d) Perth or Panama City or Pisa is located in Europe.
- Question : 8E - P, Q, and R are propositional forms, and P is equivalent to Q, and Q is equiv- alent to R. Prove that ? (a) Q is equivalent to P. (b) P is equivalent to R. (c) ?Q is equivalent to ?P.
- Question : 9E - Use a truth table to determine whether each of the following is a tautology, a contradiction, or neither. (a) (P ? Q) ? (?P ? ?Q). (b) ?(P ? ?P). ? (c) (P ? Q) ? (?P ? ?Q). (d) (A ? B) ? (A ? ?B) ? (?A ? B) ? (?A ? ?B). (e) (Q ? ?P) ? ? (P ? R). (f) P ? [(?Q ? P) ? (R ? Q)].
- Question : 10E - Suppose A is a tautology and B is a contradiction. Are the following tautolo- gies, contradictions, or neither? ? (a) A ? B. (b) A ? ?B. ? (c) A ? B. (d) ?( ?A ? B).
- Question : 11E - Give a useful denial of each statement. ? (a) x is a positive integer. (Assume that x is some fixed integer.) (b) Cleveland will win the first game or the second game. ? (c) 5 ? 3. (d) 641,371 is a composite integer. ? (e) Roses are red and violets are blue. (f) T is not bounded or T is compact. (Assume that T is a fixed object.) (g) M is odd and one-to-one. (Assume that M is some fixed function.) (h) The function f has positive first and second derivatives at x 0 . (Assume that f is a fixed function and x 0 is a fixed real number.) (i) The function g has a relative maximum at x = 2 or x = 4 and a relative minimum at x = 3. (Assume that g is a fixed function.) (j) Neither z < s nor z ? t is true. (Assume that z, s, and t are fixed real numbers.) (k) R is transitive but not reflexive. (Assume that R is a fixed object.)
- Question : 12E - Restore parentheses to these abbreviated propositional forms. (a) ??P ? ?Q ? ?S. (b) Q ? ?S ? ? (?P ? Q). (c) P ? ?Q ? ?P ? ?R ? ?P ? S. (d) ?P ? Q ? ??P ? Q ? R.
- Question : 13E - Other logical connectives between two propositions P and Q are possible. (a) The word or is used in two different ways in English. We have presented the truth table for

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