To prove: That q → ( p → q ) is a tautology.

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Answer 1

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Chapter 1.1, Problem 6E

(a)

To determine

To prove: That q→(p→q) is a tautology.

(b)

To determine

To prove: That [p∧q]∧[(¬p)∨(¬q)] is a contradiction.

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Problem 11 (a) A tank is discharging water through an orifice at a depth of T meter below the surface of the water whose area is A m². The following are the values of a for the corresponding values of A: A 1.257 1.390 x 1.50 1.65 1.520 1.650 1.809 1.962 2.123 2.295 2.462|2.650 1.80 1.95 2.10 2.25 2.40 2.55 2.70 2.85 Using the formula -3.0 (0.018)T = dx. calculate T, the time in seconds for the level of the water to drop from 3.0 m to 1.5 m above the orifice. (b) The velocity of a train which starts from rest is given by the fol- lowing table, the time being reckoned in minutes from the start and the speed in km/hour: | † (minutes) |2|4 6 8 10 12 14 16 18 20 v (km/hr) 16 28.8 40 46.4 51.2 32.0 17.6 8 3.2 0 Estimate approximately the total distance ran in 20 minutes.

- Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p − 1)/2 multiple of n, i.e. n mod p, 2n mod p, ..., p-1 2 -n mod p. Let T be the subset of S consisting of those residues which exceed p/2. Find the set T, and hence compute the Legendre symbol (7|23). 23 32 how come? The first 11 multiples of 7 reduced mod 23 are 7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8. The set T is the subset of these residues exceeding So T = {12, 14, 17, 19, 21}. By Gauss' lemma (Apostol Theorem 9.6), (7|23) = (−1)|T| = (−1)5 = −1.

Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p-1)/2 multiple of n, i.e. n mod p, 2n mod p, ..., 2 p-1 -n mod p. Let T be the subset of S consisting of those residues which exceed p/2. Find the set T, and hence compute the Legendre symbol (7|23). The first 11 multiples of 7 reduced mod 23 are 7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8. 23 The set T is the subset of these residues exceeding 2° So T = {12, 14, 17, 19, 21}. By Gauss' lemma (Apostol Theorem 9.6), (7|23) = (−1)|T| = (−1)5 = −1. how come?

Answer 2

Question

Chapter 1.1, Problem 6E

(a)

To determine

To prove: That q→(p→q) is a tautology.

(b)

To determine

To prove: That [p∧q]∧[(¬p)∨(¬q)] is a contradiction.

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Answer 3

Question

Chapter 1.1, Problem 6E

(a)

To determine

To prove: That q→(p→q) is a tautology.

(b)

To determine

To prove: That [p∧q]∧[(¬p)∨(¬q)] is a contradiction.

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Answer 4

Question

Chapter 1.1, Problem 6E

(a)

To determine

To prove: That q→(p→q) is a tautology.

(b)

To determine

To prove: That [p∧q]∧[(¬p)∨(¬q)] is a contradiction.

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Answer 5

Chapter 1.1, Problem 6E

(a)

To determine

To prove: That q→(p→q) is a tautology.

(b)

To determine

To prove: That [p∧q]∧[(¬p)∨(¬q)] is a contradiction.

Answer 6

Chapter 1.1, Problem 6E

(a)

To determine

To prove: That q→(p→q) is a tautology.

Answer 7

Chapter 1.1, Problem 6E

(a)

To determine

To prove: That q→(p→q) is a tautology.

Answer 8

(b)

To determine

To prove: That [p∧q]∧[(¬p)∨(¬q)] is a contradiction.

Answer 9

(b)

To determine

To prove: That [p∧q]∧[(¬p)∨(¬q)] is a contradiction.

Answer 10

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Answer 11

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Answer 12

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Answer 13

Ch. 1.1 - True/False Questions “” means “” Ch. 1.1 - A truth table based on four simple statements...Ch. 1.1 - True/False Questions 2. If is true, then is also...Ch. 1.1 - If p and q are both false, the truth value of...Ch. 1.1 - If pq is false, the truth value of (pq)(pq) is...Ch. 1.1 - pq andqp are logically equivalent.Ch. 1.1 - True/False Questions 7. A statement and its...Ch. 1.1 - (pq)(pq) is a tautology.Ch. 1.1 - True/False Questions 9. If B is a tautology and A...Ch. 1.1 - True/False Questions 10. If A and B are both...

To prove: That q → ( p → q ) is a tautology.

Solution Summary: The author explains that the statement qto (pt) is a tautology. The truth table for the compound statement is drawn.

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To prove: That q→(p→q) is a tautology.

To prove: That [p∧q]∧[(¬p)∨(¬q)] is a contradiction.

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