Discrete Mathematics - Chapter 4: Elementary number theory and method of proof
A real number r is irrational only if it ...... be expressed as a quotient of two integers with a nonzero denominator.
can't
An integer d doesn't divide n only if ....... (∀ integers k, n ≠ ......)
n/d is not an integer, d * k
"Zero product property": If neither of two real numbers in zero, then their product is also .......
not zero
Any two consecutive integers have opposite .......
parity
"Method of Direct Proof": 1. Express the statement to be proved in the form "∀x ∈ D, if P(x) then Q(x)." (This step is often done mentally.) 2. Start the proof by supposing x is a ...... of D for which the hypothesis P(x) is ....... (This step is often abbreviated "Suppose x ∈ D and P(x).") 3. Show that the conclusion Q(x) is true by using ........., previously established results, and the rules for logical inference.
particular but arbitrarily chosen element, true, definitions
The double of a rational number is rational or irrational?
rational
The square of any odd integer has the form .......... for some integer m.
8m + 1
"The Quotient-Remainder Theorem": Given any integer n and positive integer d, there exist unique integers q and r such that n= d * q + r and .... ≤ r < .......
0, d
"Divisibility by a prime": Any integer n > ..... is divisible by a prime number
1
The only divisors of 1 are ......... and .........
1, -1
n is composite only if ∃ positive integers r and s, such that n = r * s and ..... < r < n, ..... < s < n
1, 1
Filling the blank for equaling the statement: n is divisible by d: 1. ....... is a multiple of ........ 2. ....... is a factor of ........ 3. ....... is divisor of ........ 4. ....... divides .......
1. n, d 2. d, n 3. d, n 4. d, n
The form of a rational r: ∃ integers a & b such that r = ..... and b ≠ .....
a/b, 0
Suppose m is an integer. If m mod 11 = 6, what is 4m mod 11?
2
"3,300 in standard factored form": 3,300 = 100 * 33 = 4 * 25 * 3 *11 = 2 * 2 * 5 * 5 * 3 * 11 = 2ˆ.. * 3ˆ... * 5ˆ.... * 11ˆ....
2, 1, 2, 1
List of first 10 prime numbers: .......
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
n is even only if ∃ an integer k such that n = .....
2k
n is odd only if ∃ an integer k such that n = ......
2k + 1
List of first 10 composite numbers: .......
4, 6, 8, 9, 10, 12, 14, 15, 16, 18
"Suppose ∃x in D such that P(x) and ∼Q(x). Then P(x) and ∼P(x)" is Proof by ...........
Contradiction
"Suppose x is an arbitrary element of D such that ∼Q(x). Then ∼P(x)" is Proof by ..........
Contraposition
For all integers a and b, if a and b are positive and a divides b, then ..... ≤ ......
a, b
Does 4 | 15? No, 15/4 = 3.75, which is not .......
an integer
What is "the parity property" of any integer?
any integer is either even or odd
"Method of Proof by Contraposition" 1. Express the statement to be proved in the form ∀x in D, if P(x) then Q(x). (This step may be done mentally.) 2. Rewrite this statement in the....... form ∀x in D, if Q(x) is false then P(x) is false. (This step may also be done mentally.) 3. Prove the contrapositive by a direct proof. a. Suppose x is a ....... element of D such that Q(x) is ...... b. Show that ....... is false.
contrapositive, particular but arbitrarily chosen, false, P(x)
d | n ≡ ∃ an integer k such that n = .........
d * k
For all integers n, if n² is even then n is ........
even
The parity of an integer refers to whether the integer is ...... or .....
even, odd
"Method of Generalizing from the Generic Particular": To show that ...... element of a set ........ a certain property, suppose x is a ........ element of the set, and show that x satisfies the property.
every, satisfies, particular but arbitrarily chosen,
Given any integer n and positive integer d, there exist unique integers q and r such that ....... (The quotient-remainder theorem - chapter 4.4)
n = d * q + r and 0 ≤ r < d.
True or False? "There is a greatest integer."
false
True or False? "There is an integer that is both even and odd."
false
"Method of Proof by Contradiction" 1. Suppose the statement to be proved is ....... That is, suppose that the ...... of the statement is true. 2. Show that this supposition leads logically to a .......... 3. Conclude that the statement to be proved is true.
false, negation, contradiction
Given an integer n and a positive integer d, 1. n div d = the ........ obtained when n is divided by d, and 2. n mod d = the ...... obtained when n is divided by d.
integer quotient, nonnegative integer remainder
The sum of any rational number and any irrational number is rational or irrational?
irrational
Given any integer n > 1, the standard factored form of n is an expression of the form: n = p₁ˆe₁ * p₂ˆe₂ * .... * pκˆeκ where k is a "......."; p1, p2,..., pk are "......"; e1,e2,...,ek are "......"; and p₁ < p₂ < ··· < pκ.
positive integer, prime numbers, positive integers
"A positive divisor of a positive integer": For all integers a and b, if a and b are ...... and a ....... b, then a ≤ b (a ≤ k * a ≤ ....)
positive, divides, b
if n and d are integers and d > 0, then n div d = ..... and n mod d = .... ⇔ n = d * q + r where q and r are integers and ..... ≤ r < ......
q, r, 0, d
A real number r is rational only if it can be expressed as a ....... of two ....... with a ...... denominator.
quotient, integers, nonzero
n is prime only if ∀ positive integers r and s, if n = ........ then either r = 1 and s = n or r = n and then s = 1
r * s
The sum of any two rational number is rational or irrational?
rational
For all integers a, b, and c, if a divides b and b divides c, then a divides c. b = a * k c = b * m c = a * (k * m) What kind of rule of 3 integers a, b, c above?
transitivity
True of False? "There is no greatest integer"
true
True or False? " For all real numbers r, | − r| = |r|."
true
True or False? "Every integer is a rational number".
true
True or False? "The sum of any two even integers is even."
true
True or False? "There is no integer that is both even and odd."
true
For all real numbers x and y, |x + y| ≤ |x| + |y| or |x + y| ≥ |x| + |y|?
|x + y| ≤ |x| + |y|
For all real numbers r, what is the relation among r, -|r|, and |r|?
−|r| ≤ r ≤ |r|
