Exercise 1.1.2: Expressing English sentences using logical notation.
Express each English statement using logical operations ∨, ∧, ¬ and the propositional variables t, n, and m defined below. The use of the word "or" means inclusive or.
t: The patient took the medication.
n: The patient had nausea.
m: The patient had migraines.
(a)The patient had nausea and migraines.
(b)The patient took the medication, but still had migraines.
(c)The patient had nausea or migraines.
(d)The patient did not have migraines.
(e)Despite the fact that the patient took the medication, the patient had nausea.
(f)There is no way that the patient took the medication.
Exercise 1.2.1: Truth values for compound English sentences.
Determine whether the following propositions are true or false:
(a)5 is an odd number and 3 is a negative number.
(b)5 is an odd number or 3 is a negative number.
(c)8 is an odd number or 4 is not an odd number.
(d)6 is an even number and 7 is odd or negative.
(e)It is not true that either 7 is an odd number or 8 is an even number (or both
Exercise 1.3.2: The inverse, converse, and contrapositive of conditional sentences in English.
Give the inverse, contrapositive, and converse for each of the following statements:
(a)If she finished her homework, then she went to the party.
(b)If he trained for the race, then he finished the race.
(c)If the patient took the medicine, then she had side effects.
(d)If it was sunny, then the game was held.
(e)If it snowed last night, then school will be cancelled
Exercise 1.4.2: Truth tables to prove logical equivalence.
Use truth tables to show that the following pairs of expressions are logically equivalent.
(a)
p ↔ q and (p → q) ∧ (q → p)
(b)
¬(p ↔ q) and ¬p ↔ q
(c)
¬p → q and p ∨ q
Exercise 1.5.1: Label the steps in a proof of logical equivalence.
Below are several proofs showing that two logical expressions are logically equivalent. Label the steps in each proof with the law used to obtain each proposition from the previous proposition. The first line in the proof does not have a label.
(a)
(p → q) ∧ (q ∨ p)
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(¬p ∨ q) ∧ (q ∨ p)
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(q ∨ ¬p) ∧ (q ∨ p)
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q ∨ (¬p ∧ p)
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q ∨ (p ∧ ¬p)
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q ∨ F
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q
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(b)
(¬p ∨ q) → (p ∧ q)
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¬(¬p ∨ q) ∨ (p ∧ q)
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(¬¬p ∧ ¬q) ∨ (p ∧ q)
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(p ∧ ¬q) ∨ (p ∧ q)
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p ∧ (¬q ∨ q)
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p ∧ T
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p
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(c)
r ∨ (¬r → p)
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r ∨ (¬¬r ∨ p)
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r ∨ (r ∨ p)
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(r ∨ r) ∨ p
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r ∨ p
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Exercise 2.1.1: Even and odd integers.
Indicate whether each integer n is even or odd. If n is even, show that n equals 2k, for some integer k. If n is odd, show that n equals 2k+1, for some integer k.
(a)n = -1
(b)n = -101
(c)n = 258
(d)n = 1
Exercise 2.2.3: Find a counterexample.
Find a counterexample to show that each of the statements is false.
(a)Every month of the year has 30 or 31 days.
(b)If n is an integer and n2 is divisible by 4, then n is divisible by 4.
(c)For every positive integer x, x3
(d)Every positive integer can be expressed as the sum of the squares of two integers.
(e)The multiplicative inverse of a real number x, is a real number y such that xy = 1. Every real number has a multiplicative inverse.
Exercise 2.4.1: Proving statements about odd and even integers with direct proofs.
Each statement below involves odd and even integers. An odd integer is an integer that can be expressed as 2k+1 where k is an integer. An even integer is an integer that can be expressed as 2k, where k is an integer.
Prove each of the following statements using a direct proof.
(a)The sum of an odd and an even integer is odd.
(b)The sum of two odd integers is an even integer.
(c)The square of an odd integer is an odd integer.
(d)The product of two odd integers is an odd integer.
(e)If x is an even integer and y is an odd integer, then x2+y2 is odd.
(f)If x is an even integer and y is an odd integer, then 3x+2y is even.
(g)If x is an even integer and y is an odd integer, then 2x+3y is odd.
(h)The negative of an odd integer is also odd.
(i)If x is an even integer then (−1)x=1
.(j)If x is an odd integer then (−1)x=−1
Exercise 2.5.1: Proof by contrapositive of statements about odd and even integers.
Prove each statement by contrapositive
(a) For every integer n, if n2 is odd, then n is odd.
(b) For every integer n, if n3 is even, then n is even.
(c For every integer n, if 5n+3 is even, then n is odd.
(d)For every integer n, if n2−2n+7is even, then is odd.
(e) For every integer n, if n2 is not divisible by 4, then n is odd.
(f)For every pair of integers x and y, if xy is even, then x is even or y is even.
(g)For every pair of integers x and y, if x−y is odd, then x is odd or y is odd.
(h)If n is an integer such that n ≥ 3 and 2n-1 is prime, then n is odd.
Exercise 2.6.1: Rational and irrational numbers.
You can use the fact that √2 is irrational to answer the questions below. You can also use other facts proven within this exercise.
(a)Prove that √2 /2 is irrational.
(b)Prove that 2 − √2 is irrational.
(c)Is it true that the sum of two positive irrational numbers is also irrational? Prove your answer.
(d)Is it true that the product of two irrational numbers is also irrational? Prove your answer.
(e)Is the following statement true? Prove your answer.
If x is a non-zero rational number and y is an irrational number, then y/x is irrational.