18. Mathematicians say that "statement P is stronger than statement Q" if Q is true whenever P is true, but not conversely. (In other words, "P is stronger than Q" means that P→ Q is always true, but Q→ P is not true, in general.) Use truth tables to show the following. (a) a^ b is stronger than a. (b) a is stronger than a v b. (c) a^ b is stronger than a v b. (d) b is stronger than a → → b.
18. Mathematicians say that "statement P is stronger than statement Q" if Q is true whenever P is true, but not conversely. (In other words, "P is stronger than Q" means that P→ Q is always true, but Q→ P is not true, in general.) Use truth tables to show the following. (a) a^ b is stronger than a. (b) a is stronger than a v b. (c) a^ b is stronger than a v b. (d) b is stronger than a → → b.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
question 18
Transcribed Image Text:14. Let the following statements be given.
(a) Use connectives to translate the following statement into formal logic.
If Andy is hungry and the refrigerator is empty, then Andy is mad.
p="Andy is hungry."
q = "The refrigerator is empty."
r = "Andy is mad."
(b) Construct a truth table for the statement in part (a).
(c) Suppose that the statement given in part (a) is true, and suppose also that Andy is not mad and the
refrigerator is empty. Is Andy hungry? Explain how to justify your answer using the truth table.
15. Let A be the statement p→ (q^ r). Let B be the statement q → r.
(a) Construct truth tables for A and B.
(b) Suppose statements A and B are both true. What can you conclude about statement p? Explain your answer
using the truth table.
16. Use truth tables to prove the following distributive properties for propositional logic.
(a) p^ (q v r) is logically equivalent to (p ^ q) v (p^r).
(b) pv (q^r) is logically equivalent to (pv q) ^ (pvr).
17. Use truth tables to prove the associative properties for propositional logic.
(a) pv (q vr) is logically equivalent to (pv q) v r.
(b) p^ (q^r) is logically equivalent to (p ^ q) ^r.
18. Mathematicians say that "statement P is stronger than statement Q" if Q is true whenever P is true, but not
conversely. (In other words, "P is stronger than Q" means that P→ Q is always true, but Q→ P is not true, in
general.) Use truth tables to show the following.
(a) a ^ b is stronger than a.
(b) a is stronger than a v b.
(c) a^ b is stronger than av b.
(d) b is stronger than a → b.
19. Suppose Q is a quadrilateral. Which statement is stronger? Explain.
• Q is a square.
• Q is a rectangle.
20. Which statement is stronger? Explain.
. Manchester United is the best football team in England.
•Manchester United is the best football team in Europe.
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