Discreet Math Week 2

Consider the following theorem.

Theorem. Let x be a quagrel. If x has been schlumpfed, then x is a domel.

Give the contrapositive of this theorem.

Let x be a quagrel. If x is not a domel, then x has not been schlumpfed. Let x be a domel. If x is not a quagrel, then x has been schlumpfed.     Let x be a quagrel. If x has been schlumpfed, then x is a not domel. Let x be a domel. If x is not a quagrel, then x has not been schlumpfed. Let x be a domel. If x has been schlumpfed, then x is a not quagrel.

Find a counterexample for the statement.

If p is prime, then

p2 + 4

is prime.

p = (No Response)

Consider the following theorem.

Theorem. Let x be a wamel. If x has been dorfelled, then x is a jaggleswoggle.

Give the converse of this theorem.

Let x be jaggleswoggle. If x is a wamel, then x has not been dorfelled. Let x be wamel. If x is a jaggleswoggle, then x has been dorfelled.     Let x be a jaggleswoggle. If x has been dorfelled, then x is a wamel. Let x be jaggleswoggle. If x is a wamel, then x has been dorfelled. Let x be a wamel. If x has been dorfelled, then x is not a jaggleswoggle.

Find a counterexample for the statement.

For every real number N > 0, there is some real number x such that

Nx > x.

N = (No Response)

Let the following statements be given.

Definition. A triangle is scalene if all of its sides have different lengths. Theorem. A triangle is scalene if it is a right triangle that is not isosceles.

Suppose ΔABC is a scalene triangle. Which of the following conclusions are valid? (Select all that apply.)

All of the sides of ΔABC have different lengths. ΔABC is a right triangle that is not isosceles. Neither conclusion in valid. There is not enough information.

Why or why not? Explain your reasoning.

(No Response)

This answer has not been graded yet.

Prove:

For all integers n, if n2 is odd, then n is odd.

Use a proof by contraposition, as in Lemma 1.1.

Let n be an integer. Suppose that n is even, i.e., n = (No Response) for some integer k. Then n2 = (No Response) = 2

(No Response)

 is also even.

Consider the following definition.

Definition. An integer n is sane if

3 | (n2 + 2n).

Give a counterexample to the following: All odd integers are sane. n = (No Response)

Consider the following definitions.

Definition. An integer n is alphic if

n = 4k + 1

for some integer k. Definition. An integer n is gammic if

n = 4k + 3

for some integer k.

Show that 35 is gammic.

35 = 4

(No Response)

 + 3

Consider the following model for four-point geometry.

A point "is on" the line if the line's box contains the point. Give a pair of parallel lines in this model. (Refer to Definition 1.8. Select all that apply.)

Consider the following Badda-Bing axiomatic system and definition.

Undefined terms: badda, bing, hit Axioms: 1. Every badda hits exactly four bings. 2. Every bing is hit by exactly two baddas. 3. If x and y are distinct baddas, each hitting bing q, then there are no other bings hit by both x and y. 4. There is at least one bing. Definition. Let x and y be distinct baddas. We say that a bing q is a boom of x and y, if x hits q and y hits q.

Rewrite Axiom 3 using this definition.

Two distinct baddas can have at most one boom. Two distinct booms can't have more than one badda.     Two distinct baddas can have at least one boom. One boom cannot have more than 2 distinct baddas. One boom cannot have two distinct baddas.