DDM Chapter 2

True.

(T/F) Every proof begins with the word "Proof: " and ends with symbol ■.

True.

(T/F) Universal domains may refer to more than one object which may come from different domains.

Proof by Contradiction

Proof that starts by assuming that the theorem is false and then shows that some logical inconsistency arises as a result of this assumption.

Proofs by contradiction

__________ ______ ____________ can be used to prove theorems that are not conditional statements.

Rational Number

A number that can be expressed as the ratio of two integers in which the denominator is non-zero.

false

A proof by contradiction assumes that the entire statement is __________.

Irrational Number

A real number that is not rational, e.g. pi.

Proof

A series of steps, each of which follows logically from assumptions, or from previously proven statements, whose final step should result in the statement of the theorem being proven.

Theorem

A statement that can be proven to be true.

Proof by Exhaustion

A type of proof in which if the domain of a universal statement is small, it may be easiest to prove the statement by checking each element individually.

Counterexample

An assignment of values to variables that shows that a universal statement is false.

Odd Integer

An integer that can be expressed as 2k + 1 for some integer k.

Even Integer

An integer that can be expressed as 2k for some integer k.

Generic Object

An object in which nothing is assumed about it other than the assumptions that are given in the statement of the theorem.

Parity

An unknown number that is odd or even.

3, 1, 4, and 2.

Below are the steps of a direct proof of the following theorem: Theorem: If x - 3 = 0, then x2 - 2x - 3 = 0. Put the steps of the proof in the correct order so that each step follows from previous steps in the proof. 1) Add 3 to both sides of the equation: x = 3. 2) x2 - 2x - 3 = 32 - 2⋅3 - 3 = 9 - 6 - 3 = 0. 3) Assume x - 3 = 0. 4) Plug in x = 3 into the expression x2 - 2x - 3.

A, B, B, B, C.

Below is a statement of a theorem and a proof by cases with some parts of the argument replaced by capital letters in red font. Theorem: Every perfect square is either a multiple of 4 or a multiple of 4 plus 1. Proof. Every perfect square can be expressed as n2 for some integer n. We consider two cases: Case 1: n is even. If n is even, then it can be expressed as *A*, for some integer k. Plug in the expression for n into n2: n2=(*A*)2=*B* n2=(*A*)2=*B* Since k2 is an integer, if n is even, then n2 is a multiple of 4. Case 2: n is *C*. If n is *C*, then n can be expressed as *D*, for some integer k. Plug in the expression for n into n2: n2=(*D*)2=*E* n2=(*D*)2=*E* Since (k2+k) is an integer, if n is *C*, then n2 is one plus a multiple of 4. ■ What is the correct expression for *A*? A) 2k B) 2k+1 C) even D) 2 What is the correct expression for *B*? A) 2k+1 B) 4k^2 C) multiple of 4 D) 3 What is the correct expression for *C*? A) even B) odd C) a multiple of 4 D) 4 What is the correct expression for *D*? A) 2k B) 2k+1 C) odd D) 5 What is the best expression for *E*? Note that more than one of the choices may make the equality true. However, the goal is to express n2 as a multiple of 4 or one plus a multiple of 4. A) 4k^2 + 2k +1 B) (2k)^2 + 4k + 1. C) 4(k^2+k) + 1

A, B, C, and B.

Consider the following statement: For every real number x, if 0 ≤ x ≤ 3, then 15 - 8x + x^2 > 0 1) What would be the starting assumption in a direct proof of the statement above? A) 0 ≤ x ≤ 3 B) 15-8x+x^2 > 0 C) 15-8x+x^2 ≤ 0 2) What would be proven in a direct proof of the statement, given the assumption? A) 0 ≤ x ≤ 3 B) 15-8x+x^2 > 0 C) 15-8x+x^2 ≤ 0 3) What would be the starting assumption in a proof by contrapositive of the statement above? A) 0 ≤ x ≤ 3 B) 15-8x+x2 > 0 C) 15-8x+x2 ≤ 0 What would be proven in a proof by contrapositive of the statement, given the assumption? A) 0 ≤ x ≤ 3 B) x < 0 or x > 3. C) x < 0 and x > 3.

Absolute Value

Defined to be |x| = -x if x ≤ 0, and |x| = x if x ≥ 0..

false; inconsistency

In proof by contradiction, assume the theorem is ________ and show _________.

C, A, B.

Match the type of the proof to its logical format. In each case the theorem being proved is p→ q. 1) Assume p. Follow a series of steps to conclude q. A) Proof by contrapositive B) Proof by contradiction C) Direct proof 2) Assume ¬q. Follow a series of steps to conclude ¬p. A) Proof by contrapositive B) Proof by contradiction C) Direct proof 3) Assume p ∧ ¬q. Follow a series of logical steps to conclude r ∧ ¬r for some proposition. A) Proof by contrapositive B) Proof by contradiction C) Direct proof

Proof by Cases

Proof of a universal statement such as ∀x P(x) that breaks the domain for the variable x into different classes and gives a different proof for each class.

Proof by Contrapositive

Proves a conditional theorem of the form p → c by showing that the contrapositive ¬c → ¬p is true.

Axioms

Statements assumed to be true.

Direct Proof

The hypothesis p is assumed to be true and the conclusion c is proven as a direct result of the assumption.

biconditional

Theorems are expressed as ____________ statements.

hypothesis

Universal quantifier and domain are expressed as part of the ____________, e.g. "If n is an odd integer, then n^2 is an odd integer."

Name a generic object in the domain.

What is the first step in proving a universal statement?

Assume that the hypothesis is true.

What is the first step of direct proof?

Assume that the conclusion is false.

What is the first step of proof by contrapositive?

Show that the hypothesis is false.

What is the last step of proof by contrapositive?

Write down anything you know about the object and what needs to be proven.

What is the second step in proving a universal statement?