Math Answers

A cadet rappelling down a cliff on a rope needs help. A cadet on the ground pulls tight on the end of the rope that hangs down from the rappelling cadet to lock the cadet in place. The length of the rope between the two cadets is 129 feet, and the angle of elevation of the rope is 75°. How high above the ground is the cadet on the rope? (Round your answer to two decimal places.)

Solve for x when a,b and c are negative constants

1. (ax+b)/c≤b

2. If a>1 then prove that a²>1

 Show that ∃xP(x) ∧ ∃xQ(x) and ∃x(P(x) ∧ Q(x)) are not logically equivalent.

 Show that ∃x(P(x) ∨ Q(x)) and ∃xP(x) ∨ ∃xQ(x) are logically equivalent.

 Determine whether ∀x(P(x) ↔ Q(x)) and ∀x P(x) ↔ ∀xQ(x) are logically equivalent. Justify your answer.

 Determine whether ∀x(P(x) → Q(x)) and ∀xP(x) → ∀xQ(x) are logically equivalent. Justify your answer.

 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) Some old dogs can learn new tricks. b) No rabbit knows calculus. c) Every bird can fly. d) There is no dog that can talk. e) There is no one in this class who knows French and Russian.

 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 horse that can add. c) Every koala can climb. d) No monkey can speak French. e) There exists a pig that can swim and catch fish.

 Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives.

a) No one is perfect. b) Not everyone is perfect. c) All your friends are perfect. d) At least one of your friends is perfect. e) Everyone is your friend and is perfect. f ) Not everybody is your friend or someone is not perfect.

 Let C(x) be the statement “x has a cat,” let D(x) be the statement “x has a dog,” and let F(x) be the statement “x has a ferret.” Express each of these statements in terms of C(x), D(x), F(x), quantifiers, and logical connectives. Let the domain consist of all students in your class. a) A student in your class has a cat, a dog, and a ferret. b) All students in your class have a cat, a dog, or a ferret. c) Some student in your class has a cat and a ferret, but not a dog. d) No student in your class has a cat, a dog, and a ferret. e) For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet.