Exam 2

Proofs should always be written in _____________sentences, and each assertion made in a proof should be accompanied by a ______________________________________________.

complete reason that justifies the assertion

Given any real number x, the ceiling of x is the unique integer n such that _____________________________________________________________________.

n - 1 < x ≤ n

Given any real number x, the floor of x is the unique integer n such that__________________________________________________________________.

n ≤ x < n + 1

In general, a statement of the form "∀x ∈ D, if P(x) then Q(x)" is called vacuously true if, and only if, _____________________,

P(x) is false for every x in D

Disprove the following statement: if the sum of two integers is even, then one of the integers is even.

Proof: Counterexample Let a = 5 and b = 7 [both are odd] a + b = 12 [a + b is even] Hence, the statement is false.

Use proof by the method of exhaustion: For each integer n with 1 ≤ n ≤ 10, n2 - n + 11 is a prime number.

Proof: Show that n2 - n + 11 is prime for each individual n in D. 12 - 1 - 11 = 11 which is prime 22 - 2 - 11 = 13 which is prime 32 - 3 - 11 = 17 which is prime 42 - 4 - 11 = 23 which is prime 52 - 5 - 11 = 31 which is prime 62 - 6 - 11 = 41 which is prime 72 - 7 - 11 = 51 which is prime 82 - 8 - 11 = 67 which is prime 92 - 9 - 11 = 83 which is prime 102 - 10 - 11 = 101 which is prime Hence, the statement is true.

Prove the following statement: the product of any even integer and any integer is even.

Proof: Suppose a is any [PBAC] even integer and that b is any [PBAC] integer [we need to show that the product of a and b is even]. By definition of even, a = 2k for some integer k. By substitution, ab = 2km Let t = k m. Then, t is an integer because the product of integers is an integer. By substitution, ab = 2t Hence, by definition of even, ab is even [as was to be shown].

Using proof by contraposition, show that if n2 is even then n is even.

Proof: Suppose n is any odd integer. [We must show that n2 is odd.] By definition of odd, n = 2k + 1 for some integer k. By substitution and algebra, n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. But 2k2 + 2k is an integer because products and sums of integers are integers. So n2 = 2(an integer) + 1, and thus, by definition of odd, n2 is odd [as was to be shown].

Using proof by contradiction, show that there is no greatest even integer.

Proof: [We take the negation of the statement and suppose it to be true.] Suppose not. That is, suppose there is a greatest even integer N. [We must deduce a contradiction.] Then N ≥ n for every even integer n. Let M = N + 2. Now M is an integer since it is a sum of integers and it is even because N = 2k for some integer K by definition of even. Therefore, by substitution and algebra, M = 2k + 2 = 2(k + 1). Also M > N since M = N + 2. Thus M is an even integer that is greater than N. So N is the greatest even integer and N is not the greatest even integer, which is a contradiction. [This contradiction shows that the supposition is false and, hence, that the statement is true.]

A statement of the form "∀x ∈ D, Q(x)" is true if, and only if,

Q(x) is true for each individual x in D.

Given integers a and b, if there exists an integer k such that b = ak, then ______________________________________________________________________.

a divides b (or a | b, or a is a factor of b; or a is a divisor of b; or b is divisible by a; or b is a multiple of a)

A predicate is

a sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables.

A student gives the following INCORRECT proof on an exam. Statement: The sum of any two even integers equals 4k for some integer k. Proof: Suppose m and n are any two [PBAC] even integers. By definition of even, m = 2k for some integer k and n = 2k for some integer k. By substitution, m + n = 2k + 2k = 4k[as was to be shown] a. Explain the mistake. b. Give a correct direct proof.

a. The proof is not properly written and it uses k to represent two different amounts. b. Proof: Suppose m and n are any two [PBAC] even integers. [We must show that m + n is even.] By definition of even, m = 2k for some integer k and n = 2x for some integer x. By substitution and algebra, m + n = 2k + 2x = 2(k + x) Let t = k + x, t is an integer because it is sum of integers. By substitution, m + n = 2t Hence, by definition of even, m + n is even[as was to be shown].

To find a counterexample for a statement of the form "∀x in D, if P(x) then Q(x)" you find ___________________________________________________________________.

an element of D for which P(x) is true and Q(x) is false

If you want to establish the truth of a statement of the form "∃x ∈ D such that ∀y ∈ E, P(x, y)," your job is to find ____________________ with the property that no matter what ____________________, P(x, y) will be ____________________.

an element x in D element y in E anyone might choose true

An integer is even if, and only if, it ________________________________________.

equals twice some integer

An integer is odd if, and only if, it _________________________________________.

equals twice some integer plus 1

A negation of a universal statement is an ____________________statement.

existential

A real number is rational if, and only if, ___________________________________.

it can be written as a ratio of integers with a nonzero denominator

An integer is prime if, and only if, _________________________________________.

it is greater than 1, and if it is written as a product of positive integers, then one of the integers is 1

An integer is composite if, and only if, _____________________________________.

it is greater than 1, and it can be written as a product of positive integers neither of which is 1

If a and b are integers, the notation a | b stands for ___________________________, and the notation a/b stands for ______________________________________________.

the sentence "a divides b" the real number a divided by b (if b ≠ 0)

The truth set of a predicate P(x) with domain D is

the set of all x in D such that P(x) is true.

To prove a statement by contradiction, you suppose that _________________ and you show that _____________________________________________________________________.

the statement is false this supposition leads to a contradiction

An integer a divides an integer b if, and only if,_____________________________ .

there is an integer, say k, such that b = ak

A statement of the form "∃x ∈ D, Q(x)" is true if, and only if,

there is at least one x in D for which Q(x) is true.

The fact that a universal statement is true in some instances does not imply that it is ___________________________________.

true in all instances

When writing a proof, it is a mistake to use the same letter to represent ______________________________________________________________________.

two different quantities

A negation of an existential statement is a ____________________statement.

universal

If you want to establish the truth of a statement of the form "∀x ∈ D, ∃y ∈ E such that P(x, y)," your challenge is to allow someone else to pick ____________________, and then you must find ____________________ for which P(x, y) ____________________.

whatever element x in D they wish an element y in E is true

According to the method of direct proof, to prove that a statement of the form "x in D, if P(x) then Q(x)" is true, you suppose that _________________________ and you show that __________________________________________.

x is any [particular but arbitrarily chosen] element of D for which P(x) is true Q(x) is true

To prove a statement of the form "x in D, if P(x) then Q(x)" by contraposition, you suppose that _________________________________________and you show that_______________________________________.

x is any [particular but arbitrarily chosen] element of D for which Q(x) is false P(x) is false

According to the method of generalizing from the generic particular, to prove that every element of a domain satisfies a certain property, you suppose that _________________________ and you show that ______________________________.

you have a particular but arbitrarily chosen element of the domain that element satisfies the property

19. A negation for a statement of the form "∃x ∈ D such that ∀y ∈ E, P(x, y)" is ____________________.

∀x ∈ D, ∃y ∈ E such that ∽ P(x, y)

A negation for a statement of the form "∃x ∈ D such that Q(x)" is ____________________.

∀x ∈ D, ∽ Q(x)

A universal conditional statement is a statement of the form

∀x, if P(x) then Q(x), where P(x) and Q(x) are predicates.

A statement of the form "All A are B" can be written with a quantifier and a variable as ____________________.

∀x, if x is an A then x is a B

A statement of the form "No A are B" can be written with a quantifier and a variable as ____________________.

∀x, if x is an A then x is not a B (Or: ∀x, if x is an B then x is not a A)

Given a statement of the form "∀x, if P(x) then Q(x)," the contrapositive is ____________________, the converse is ____________________, and the inverse is ____________________.

∀x, if ∽Q(x) then ∽P(x), ∀x, if Q(x) then P(x), ∀x, if ∽P(x) then ∽Q(x)

A statement of the form "Some A are B" can be written with a quantifier and a variable as ____________________.

∃x such that x is an A and x is a B

A negation for a statement of the form "∀x ∈ D, if P(x) then Q(x)" is ____________________.

∃x ∈ D such that P(x) and ∽ Q(x)

A negation for a statement of the form "∀x ∈ D, ∃y ∈ E such that P(x, y)" is ____________________.

∃x ∈ D such that ∀y ∈ E, ∽ P(x, y)

A negation for a statement of the form "∀x ∈ D, Q(x)" is ____________________.

∃x ∈ D such that ∽ Q(x)

Given x = -7.34, what are ⌊x⌋ and ⌈x⌉ ?

⌊x⌋ = -8 ⌈x⌉ = -7