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is equivalent to P V R. 9. Suppose P, Q, S, and R are propositional forms, P is equivalent to Q, and Sis equivalent to R. For each pair of forms, determine whether they are necessar- ily equivalent. If they are equivalent, explain why. (a) P and R (c) (e) * * PAS and QAR ~(PAS) and~Qv~R (b) Pande (d) PV S and QVR (f) PAQ and SAR

is equivalent to P V R. 9. Suppose P, Q, S, and R are propositional forms, P is equivalent to Q, and Sis equivalent to R. For each pair of forms, determine whether they are necessar- ily equivalent. If they are equivalent, explain why. (a) P and R (c) (e) * * PAS and QAR ~(PAS) and~Qv~R (b) Pande (d) PV S and QVR (f) PAQ and SAR

Advanced Engineering Mathematics
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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In logic what is the difference between propositional forms that are necessarily equivalent and propositional forms which are equivalent? In chapter 1.1 of book “A Transition to Advanced Mathematics” by Douglas Smith, he states two propositional forms are equivalent if they have the same truth table. The book doesn’t mention anything about “necessary equivalence” until chapter 1.1, exercise 9: “Suppose P, Q, S, and R are propositional forms, P is equivalent to Q, and S is equivalent to R. For each pair of forms, determine whether they are necessarily equivalent. If they are equivalent explain why.”
-8 hot CHAPTER 1 Logic and Proofs h bag y sou (f) P is "Hexagons have 5 sides" and Q is "V2.3 = 2√3." 3. Make a truth table for each of the following propositional forms. ★ PA~P ★ ★ ★ (e) P is "0, 5, and 10 are all natural numbers" and Q is "98 has two prime divisors." (a) (c) (e) (g) (i) (k) PA~Q (PAQ) V~Q (PV~Q) AR ★ (a) QA (RAS) * (c) (e) * 5. 6. ★ (a) ★ (c) (e) PA (QVR) PAP (d) (e) (f) (g) 4. If P, Q, and R are true while S and K are false, which of the following are true? ★ (c) (e) (b) (d) (f) (h) (j) (1) (b) (d) PV~P PA (QV~Q) ~(P^Q) ~P~Q (PVQ) ^ (RVS) ~PV~Q (PVS) A (PVK) Use truth tables to verify each part of Theorem 1.1.1. Which of the following pairs of propositional forms are equivalent? (~P) V (~Q), ~(PV~Q) ~P~Q, ~(P^~Q) ~(PQ), ~P^~Q (PAQ) VP, P (b) (PAQ) VR, PA (QVR) (d) (PAQ) VR, PV (QAR) (f) (PAQ) V (PAR) (PAQ) V (R^~S) QV (RAS) (~PV~Q) V (~RV~S) (~QVS) ^ (QVS) 7. Determine the propositional form and truth value for each of the following: (a) It is not the case that gold is not a metal. (b) * (c) PA S and QAR ~(PAS) and ~Qv~R * (f) (h) KA~(SVQ) 19 and 79 are prime, but 119 is not. Julius Caesar was born in 1492 or 1493 and died in 1776. Perth or Panama City or Pisa is located in Europe. Although 51 divides 153, it is neither prime nor a divisor of 409. While the number is greater than 3, the sum 1 + 27 is less than 8. It is not the case that both -5 and 13 are elements of N, but 4 is in the set of rational numbers. 8. Suppose P, Q, and R are propositional forms. Explain why each is true. * (a) If P is equivalent to Q, then Q is equivalent to P. (b) If P is equivalent to Q, and Q is equivalent to R, then P is equivalent to R. (c) If P is equivalent to Q, then ~P is equivalent to~Q. (d) If Q is equivalent to R, then P Q is equivalent to PAR. (e) If Q is equivalent to R, then PV Q is equivalent to PV R. 9. Suppose P, Q, S, and R are propositional forms, P is equivalent to Q, and S is equivalent to R. For each pair of forms, determine whether they are necessar- ily equivalent. If they are equivalent, explain why. *(a) P and R (b) Pande (d) (f) PV S and QVR PAQ and SAR
Transcribed Image Text:-8 hot CHAPTER 1 Logic and Proofs h bag y sou (f) P is "Hexagons have 5 sides" and Q is "V2.3 = 2√3." 3. Make a truth table for each of the following propositional forms. ★ PA~P ★ ★ ★ (e) P is "0, 5, and 10 are all natural numbers" and Q is "98 has two prime divisors." (a) (c) (e) (g) (i) (k) PA~Q (PAQ) V~Q (PV~Q) AR ★ (a) QA (RAS) * (c) (e) * 5. 6. ★ (a) ★ (c) (e) PA (QVR) PAP (d) (e) (f) (g) 4. If P, Q, and R are true while S and K are false, which of the following are true? ★ (c) (e) (b) (d) (f) (h) (j) (1) (b) (d) PV~P PA (QV~Q) ~(P^Q) ~P~Q (PVQ) ^ (RVS) ~PV~Q (PVS) A (PVK) Use truth tables to verify each part of Theorem 1.1.1. Which of the following pairs of propositional forms are equivalent? (~P) V (~Q), ~(PV~Q) ~P~Q, ~(P^~Q) ~(PQ), ~P^~Q (PAQ) VP, P (b) (PAQ) VR, PA (QVR) (d) (PAQ) VR, PV (QAR) (f) (PAQ) V (PAR) (PAQ) V (R^~S) QV (RAS) (~PV~Q) V (~RV~S) (~QVS) ^ (QVS) 7. Determine the propositional form and truth value for each of the following: (a) It is not the case that gold is not a metal. (b) * (c) PA S and QAR ~(PAS) and ~Qv~R * (f) (h) KA~(SVQ) 19 and 79 are prime, but 119 is not. Julius Caesar was born in 1492 or 1493 and died in 1776. Perth or Panama City or Pisa is located in Europe. Although 51 divides 153, it is neither prime nor a divisor of 409. While the number is greater than 3, the sum 1 + 27 is less than 8. It is not the case that both -5 and 13 are elements of N, but 4 is in the set of rational numbers. 8. Suppose P, Q, and R are propositional forms. Explain why each is true. * (a) If P is equivalent to Q, then Q is equivalent to P. (b) If P is equivalent to Q, and Q is equivalent to R, then P is equivalent to R. (c) If P is equivalent to Q, then ~P is equivalent to~Q. (d) If Q is equivalent to R, then P Q is equivalent to PAR. (e) If Q is equivalent to R, then PV Q is equivalent to PV R. 9. Suppose P, Q, S, and R are propositional forms, P is equivalent to Q, and S is equivalent to R. For each pair of forms, determine whether they are necessar- ily equivalent. If they are equivalent, explain why. *(a) P and R (b) Pande (d) (f) PV S and QVR PAQ and SAR
meorem 1.1.1 1.1 Propositions and Connectives Both ~(PV~P) and Q^~Q are examples of contradictions. The negation of a contradiction is, of course, a tautology. Particularly important in writing proofs will be the ability to recognize or write a statement equivalent to another. Sometimes, knowledge of the mathematical con- tent enables us to write an equivalent statement. For instance, if one step in a proof is the statement "The ones digit of the integer x is zero," a later step could be the equivalent statement "The integer x is divisible by 10." In other cases, the meaning of a statement does not come into play; it is the form of the statement that may be used to find a useful equivalent. We say two propositional forms are equivalent if they have the same truth tables. Some of the most commonly used equivalences are presented in the follow- ing theorem. You may wish to make truth tables for each pair of forms to verify that they are equivalent, but in each case you should understand the equivalences by examining their meanings. For example, in part (h), negation is applied to a conjunction. The form ~(PAQ) is true precisely when P Q is false. This happens when one of P or Q is false, or, in other words, when one of ~P or ~Q is true. Thus, ~(PQ) is equivalent to ~PV~Q. That is, to say "We don't have both P and Q" is the same as saying "We don't have P or we don't have All parts of Theorem 1.1.1 may be verified by constructing truth tables for each pair of propositional forms. (See Exercise 5.) For propositions P, Q, and R, the following are equivalent: (a) P and (b) PVQ and (c) PAQ and PV (QVR) and PA (QAR) and PA (QVR) and PV (QAR) and (PVO)A (PVR) ~PV~Q ~P^~Q (d) (e) (h) (i) ~(P^Q) and ~(P VQ) and ~(~P) QVP QAP (PVQ) VR (PAQ) AR (PAQ) V (PAR) 5 Double Negation Law Commutative Laws Associative Laws Distributive Laws DeMorgan's Laws As an example of how this theorem might be useful, suppose that for some integer x we have determined that the statement "x is even and x > 10" is not true. Then its negation, "It is not the case that the integer x is even and x > 10," is true and has the form ~(PQ), where P is "x is even" and Q is "x> 10." By part (h) of Theorem 1.1.1, this is equivalent to ~PV~Q, which is "It is not the case that x is even or it is not the case that x > 10." * Augustus DeMorgan (1806-1871) was an English logician and mathematician whose contributions include his notational system for symbolic logic. He also introduced the term mathematical induction (see Section 2.4) and developed a rigorous foundation for that proof technique.
Transcribed Image Text:meorem 1.1.1 1.1 Propositions and Connectives Both ~(PV~P) and Q^~Q are examples of contradictions. The negation of a contradiction is, of course, a tautology. Particularly important in writing proofs will be the ability to recognize or write a statement equivalent to another. Sometimes, knowledge of the mathematical con- tent enables us to write an equivalent statement. For instance, if one step in a proof is the statement "The ones digit of the integer x is zero," a later step could be the equivalent statement "The integer x is divisible by 10." In other cases, the meaning of a statement does not come into play; it is the form of the statement that may be used to find a useful equivalent. We say two propositional forms are equivalent if they have the same truth tables. Some of the most commonly used equivalences are presented in the follow- ing theorem. You may wish to make truth tables for each pair of forms to verify that they are equivalent, but in each case you should understand the equivalences by examining their meanings. For example, in part (h), negation is applied to a conjunction. The form ~(PAQ) is true precisely when P Q is false. This happens when one of P or Q is false, or, in other words, when one of ~P or ~Q is true. Thus, ~(PQ) is equivalent to ~PV~Q. That is, to say "We don't have both P and Q" is the same as saying "We don't have P or we don't have All parts of Theorem 1.1.1 may be verified by constructing truth tables for each pair of propositional forms. (See Exercise 5.) For propositions P, Q, and R, the following are equivalent: (a) P and (b) PVQ and (c) PAQ and PV (QVR) and PA (QAR) and PA (QVR) and PV (QAR) and (PVO)A (PVR) ~PV~Q ~P^~Q (d) (e) (h) (i) ~(P^Q) and ~(P VQ) and ~(~P) QVP QAP (PVQ) VR (PAQ) AR (PAQ) V (PAR) 5 Double Negation Law Commutative Laws Associative Laws Distributive Laws DeMorgan's Laws As an example of how this theorem might be useful, suppose that for some integer x we have determined that the statement "x is even and x > 10" is not true. Then its negation, "It is not the case that the integer x is even and x > 10," is true and has the form ~(PQ), where P is "x is even" and Q is "x> 10." By part (h) of Theorem 1.1.1, this is equivalent to ~PV~Q, which is "It is not the case that x is even or it is not the case that x > 10." * Augustus DeMorgan (1806-1871) was an English logician and mathematician whose contributions include his notational system for symbolic logic. He also introduced the term mathematical induction (see Section 2.4) and developed a rigorous foundation for that proof technique.
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