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Discrete Mathematics Q&A Archive of July 20, 2021

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July 20 of 2021

Using the euclidean algorithm, find the gcd of the given integers. $$3076,1976$$
Show that a greedy algorithm that schedules talks in a lecture hall, as described in Example $7,$ by selecting at each step the talk that overlaps the fewest other talks, does not always produce an optimal schedule.
In an old puzzle attributed to Alcuin of York $(735-804),$ a farmer needs to carry a wolf, a goat, and a cabbage across a river. The farmer only has a small boat, which can carry the farmer and only one object (an animal or a vegetable). He can cross the river repeatedly. However, if the farmerā€¦
Express the negation of each of these statements in terms of quantifiers without using the negation symbol. a) $\forall x(-2<x<3)$ b) $\forall x(0 \leq x<5)$ c) $\exists x(-4 \leq x \leq 1)$ d) $\exists x(-5<x<-1)$
Express the negation of these propositions using quantifiers, and then express the negation in English. a) Some drivers do not obey the speed limit. b) All Swedish movies are serious. c) No one can keep a secret. d) There is someone in this class who does not have a good attitude.
Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase "It is not the case that.") a) All dogs have fleas. b) There is a horseā€¦
For each of these statements find a domain for which the statement is true and a domain for which the statement is false. a) Everyone speaks Hindi. b) There is someone older than 21 years. c) Every two people have the same first name. d) Someone knows more than two other people.
Let $P(x)$ be the statement "The word $x$ contains the letter $a$ ." What are these truth values? $\begin{array}{ll}{\text { a) } P(\text { orange })} & {\text { b) } P(\text { lemon })} \\ {\text { c) } P(\text { true })} & {\text { d) } P(\text { false })}\end{array}$
Express each of these system specifications using predicates, quantifiers, and logical connectives, if necessary. a) At least one console must be accessible during every fault condition. b) The e-mail address of every user can be retrieved whenever the archive contains at least one messageā€¦
In an old puzzle attributed to Alcuin of York $(735-804),$ a farmer needs to carry a wolf, a goat, and a cabbage across a river. The farmer only has a small boat, which can carry the farmer and only one object (an animal or a vegetable). He can cross the river repeatedly. However, if the farmerā€¦
For each of these statements find a domain for which the statement is true and a domain for which the statement is false. a) Everyone speaks Hindi. b) There is someone older than 21 years. c) Every two people have the same first name. d) Someone knows more than two other people.
Let $P(x)$ be the statement "The word $x$ contains the letter $a$ ." What are these truth values? $\begin{array}{ll}{\text { a) } P(\text { orange })} & {\text { b) } P(\text { lemon })} \\ {\text { c) } P(\text { true })} & {\text { d) } P(\text { false })}\end{array}$