Discrete Mathematics Answers

h) Which assumptions are being referred to by the phrase "the assumptions in force" at a given step

in a proof?


1. Only the premises at the top level of the proof.

2. Only the assumption of the subproof in which the step-in question is embedded.

3. All of the assumptions of every subproof in which the step-in question is embedded

together with the premises.

4. Only the assumptions of every subproof in which the step-in question is embedded.

5. Only the assumptions of subproofs of equally deep nesting.




i) Which of the following domains would most clearly suggests that the sentence " 'Everyone' is a

lot of people" is not always true?


1. The set of people in the province of Gauteng.

2. The set of logicians in this video correct

3. The set of even natural numbers

4. The set of people at a well-attended concert

5. The set of points on the interval [0,1)

f) Suppose we are told that the following expression is true: P ↔ (Q ∧ ¬Q). What can we then

conclude about P's truth?


1. P must be true

2. It is uncertain whether P is true or false

3. P must be false.



g) Which of the following informal arguments is the best example of a use of conditional elimination?

In addition to being able to recognize conditional elimination you will need to use your knowledge

of the correct translation of natural language conditionals.


1. “John will prove a theorem only if he isn't very tired. He slept very well last night, so he'll

prove a theorem."

2. “John won't prove theorems if he is tired. He's pretty tired today, so he'll prove a theorem."

3. ”If Dave doesn't sleep, then he'll prove a theorem. Since he slept soundly, he

consequently proved no theorems."

4. “Dave has a good night's sleep only if he will prove a theorem. Last night, he slept quite

well and so he will prove a theorem.

Let Q+ be the set of positive rational numbers. Prove that if x is in q+ there is some y in q+ such that y < x. Provide two proofs of this fact, one using a direct proof and one using a proof by contradiction.

How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of the same suit are chosen?

If x, y are real numbers such that ordered pairs (x + y, x -y) and (2x + 3y, 3x - 2y) are equal, then (x, y) is equal to

In boolean algebra, proof: (a) x∨y = y if and only if x∧y = x

Prove: If e is an edge in a simple closed path in G, then e belongs to some cycle.

How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of the same suit are chosen?

Prove Pigeon-Hole Principle: If a finite set S is partitioned into k sets, then at least one of the sets has |S|/k or more elements.

~(p • q) → (p ∨ r ) truth table