Chapter 1: Q33E (page 109)
Adapt the proof in Example 4 in Section 1.7 to prove that if\(n = abc\), where \(a, b, and c\)are positive integers, then\(a \le \sqrt[3]{n}, b \le \sqrt[3]{n}, or c \le \sqrt[3]{n}\).
\(a \le \sqrt[3]{n},b \le \sqrt[3]{n}\)or \(c \le \sqrt[3]{n},\)is true.
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Construct a truth table for
Each inhabitant of a remote village always tells the truth or always lies. A villager will give only a “Yes” or a “No” response to a question a tourist asks. Suppose you are a tourist visiting this area and come to a fork in the road. One branch leads to the ruins you want to visit; the other branch leads deep into the jungle. A villager is standing at the fork in the road. What one question can you ask the villager to determine which branch to take?
You are eligible to be President of the U.S.A. only if you are at least 35 years old, were born in the U.S.A, or at the time of your birth both of your parents were citizens, and you have lived at least 14 years in the country. Express your answer in terms of e: “You are eligible to be President of the U.S.A.,” a: “You are at least 35 years old,” b: “You were born in the U.S.A,” p: “At the time of your birth, both of your parents where citizens,” and r: “You have lived at least 14 years in the U.S.A.”
Determine whether each of these conditional statements is true or false.
a) If, then.
b) If, then.
c) If, then.
d) If monkeys can fly, then.
Let andbe the statements “xis a professor,” “xis ignorant,” and “xis vain,” respectively. Express each of these statements using quantifiers; logical connectives; andandwhere the domain consists of all people.
a) No professors are ignorant.
b) All ignorant people are vain.
c) No professors are vain.
d) Does (c) follow from (a) and (b)?
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