Discrete Mathematics Answers

a) Which of the following is a valid inference from the sentence, "c is a small dodecahedron but isn't
either medium and left of b or right of d."

1. c is right of d
2. c is not a dodecahedron
3. c isn't either medium and left of b or right of d correct
4. c is left of b.


c) Which of the following describes the default behavior of Fitch for (a) ∧-Elim and (b) ∧-Intro,
respectively?

1. (a) fills in the first conjunct of the cited conjunction and (b) conjoins all the cited formulas
2. (a) conjoins all the cited formulas and (b) fills in the first conjunct of the cited conjunction
3. (a) conjoins the first two cited formulas and (b) results in an error
4. (a) fills in the first conjunct of the cited conjunction and (b) conjoins the first two cited
formulas.

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.

Reminders:
(h) Guidliens: 1) Find the solution an
writing the characteristic equation.
(p) 2) Find the particular solution an .
of associated linear homogenous recurrence solution by
Sabancı University
Spring 2020 MATH 204 (Homework 3)
1) Students should already know the basic definitions used in this homework from the lectures.
2) Almost all questions are taken from the textbook. Please study sections 8.2, 8.4 and lecture notes before attempting them.
(1) Find the solution of the recurrence relation an = 2an−1 + 3n with initial condition a1 = 5.
(h) (h) 3) Use initial condition a1 = 5 and then find the solution by combining an and an .
(2) Find a closed form for the generating function of the sequence an = n−1 for n = 0, 1, 2, . . . .
Guidelines: Use the basic properties of formal power series. Your answer should not contain summation. For example, writing

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