Discrete Structures - Ch 4.1, 4.4, 4.5, 5.1

N choose r

(n/r)

An integer /n/ is *prime* if, and only if...

/n/ > 1 for all positive integers /r/ and /s/. if n = rs, then either /r/ or /s/ equals /n/. (∀ positive integers r and s, if n = rs, then either r = 1 and s = n or r = n and s = 1)

An integer /n/ is *even* if, and only if...

/n/ equals twice some integer (/n/= 2/k/ for some integer /k/)

An integer /n/ is *odd* if, and only if...

/n/ equals twice some integer plus 1 (/n/= 2/k/ + 1 for some integer /k/)

Solve and justify answer: is 0 odd or even?

0 * 2 = 0 --> 0 is even

define and solve zero factorial

0! = 1

find an explicit formula for a sequence with the following initial terms: 1, -1/4, 1/9, -1/16, 1/25, -1/36, ...

1/1^2, -1/2^2, 1/3^2, -1/4^2, 1/5^2, -1/6^2

All even numbers are composite besides...

2

Solve and justify answer: is -301 even or odd?

2*-150 + 1 --> -301 is odd

compute 32 div 9

3

the quantity /n/ factorial denoted n!, is defined as n! = n * (n-1) define and solve 3!

3! = 3 * 2 * 1 = 6

solve the combination (3/2)

3!/2!(3-2)! (3 * 2 * 1) / (2 * 1 * 1) 3 / 1 = 3

convert to n = dq + r form n = 42, d = 4

42 = 4*10 + 2

compute 32 mod 9

5

define and solve 5!/2!*3!

5 * 4 * 3 * 2 * 1 / 2 * 1 * 3 * 2 * 1 5 * 4 / 2 * 1 20 / 2 = 10

Solve and justify answer: is 5 prime or composite?

5 > 1 and has no other factor other than 1 and itself --> 5 is prime

convert to n = dq + r form n = 70, d = 54

70 = 54*1 + 16

define and solve 8!/7!

8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 / 7 * 6 * 5 * 4 * 3 * 2 * 1 8 / 1 = 8

Solve and justify answer: is 8 prime or composite?

8 > 1 and has other factors other than 1 and itself (1-8, 2-4) --> 8 is composite

write 819 in standard factored form.

819 3 * 273 3^2 * 91 3^2 * 7 * 13

For all objects, /A/, /B/, and /C/... If A = B, then B = ___

A

complete the property of divisibility: if /a/ and /b/ are integers, is 3a + 3b divisible by 3?

By assumption, integers are closed under ⊕. If a ∈ Z and b ∈ Z, then a + b ∈ Z. Therefore, 3a + 3b = 3 (a + b) = 3 * (integer). 3a + 3b is divisible by 3.

For all objects, /A/, /B/, and /C/... If A = B, and B = C, then A = ___

C

what is the central concept of number theory, and also is the capacity of being divided?

Divisibility

Prove that given any 2 consecutive integers, 1 is even and the other is odd using the parity property.

Let Z = an integer, then Z = even Case 1 (Z is even): If Z is even, then Z = 2k for some k∈Z → Z + 1 = 2k + 1 for some k∈Z = odd Case 2 (Z is odd): If Z is odd, then Z= 2k + 1 for some k∈Z → Z + 1 = (2k + 1) + 1 = 2k + 2 = 2(k + 1) for some k∈Z Since integers are closed under ⊕ and k∈Z and 1∈Z → k + 1 is an integer Therefore, when Z is odd, Z + 1 = 2*integer = even

create a constructive proof of existence for the following: there exists an even integer /n/ that can be written in 2 ways as a sum of 2 prime numbers.

Since 7+3=10 and 5+5=10, 10 is an even integer that is the sum of 2 different pairs of primes, 7 & 4 and 5 & 5. Therefore, there exists an integer /n/ which /n/ = sum of 2 primes in 2 different ways.

Prove that the sum of any 2 integers is even: *The sum of any 2 even integers is even.*

Suppose /m/ and /n/ are 2 even integers. Then by the definition of an even integer, there exists /k/, and /k²/, such that m = 2k₁ and n = 2k₂ m + n = 2k₁ + 2k₂ By the distinctive property, m + n = 2(k₁ + k₂). Since the sum of 2 integers is an integer, k = k₁ + k₂ ∈ Z. m + n = k = 2 x integer Goal: Therefore, by the definition of an even integer, m + n is even.

For all objects, /A/, /B/, and /C/... T/F: A = A

True

T/F: We assume there is no integer between 0 and 1 and the set of all integers is closed under addition, subtraction, and multiplication.

True

Are sums, differences, and products of integers also integers?

Yes

for all integers /a/ and /b/, if /a/ and /b/ are positive and /a/ divides /b/, then /a/ (<,>,≤,≥) /b/.

a ≤ b

Imagine that a person decides to count his ancestors. He has 2 parents, 4 grandparents, 8 great-grandparents, and so forth. To express the pattern of the numbers, suppose that each is labeled by an integer giving its position in the row. The pattern of computed values suggest the following for each /k/:

a(k) = 2^k

let a1 = -2, a2 = -1, a3 = 0, a4 = 1, a5 = 2 compute the following: a) 5 ∑ a(k) k = 1 b) 5 ∑ k^2 k = 1

a) -2 + -1 + 0 + 1 + 2 = 0 b) 1 + 4 + 9 + 16 + 25 = 55

The letter C is used because the quantities (n/r) are also called

combinations

The notation d|n is read as...

d divides n

The notation d∤n is read as...

d does not divide n

the symbol used to represent an index of a summation is an example of a _____, because it can be replaced by any other symbol.

dummy variable

What is parity?

evenness or oddness

the product of all consecutive integers up to a given integer occurs so often in mathematics that it is given a special notation, known as _____.

factorial notation

describe existential instantiation

if you know something exists, you can give it a name

The notation am , am + 1 , am + 2 , ... , an denotes an ___

infinite sequence

what does a nonconstructive proof of existence show?

it shows either that the existence of a value fo /x/ that makes Q(x) true is guaranteed by an axiom or a previously proved theorem or that the assumption that there is no such /x/ leads to a contradiction

if one positive integer divides a second positive integer, then the first is (greater than/less than) or equal to the second. the only divisors of 1 are ___ and ___.

less than 1 and -1

suppose /m/ is an integer. if /m/ mod 11 = 6, what is 4m mod 11?

m mod 11 = 6 m = 11q + 6 4m = 4 (11q + 6) = 44q + 24 4m mod 11 4q +2 (R2) = 2

An integer /n/ is *composite* if, and only if...

n > 1 and n = rs for some integers /r/ and /s/ with 1 < r < n and 1 < s < n. (∃ positive integers r and s such that n = rs, and 1 < r < n and 1 < s < n)

what is the formula for combinations?

n!/r!(n-r)!

what is the study of properties of integers?

number theory

what does this symbol stand for: π

pi

any integer greater than 1 is (prime/composite).

prime

we typically represent a sequence as a set of elements written in a (row/column).

row

what is a function whose domain is either all the integers between 2 given integers or all the integers greater than or equal to a given integer?

sequence

how do you show the method of exhaustion

showing the truth of the predicate separately for each individual element of the domain

the k in a(k) is called a _____ or _____.

subscript or index

what does this symbol stand for: ∑ ?

summation

each individual element a(k) (read as "a sub k") is called a _____.

term

define the parity property

the fact that any integer is either even or odd

define n div d

the integer quotient obtained when /n/ is divided by /d/

what is existential generalization?

the logical principle underlying proof

define n mod d

the nonnegative integer remainder when /n/ is divided by /d/

t/f: any two consecutive have opposite parity.

true

the most comprehensive statement about the divisibility of integers is contained in the _____.

unique factorization of the integers theorem (or the fundamental theorem of arithmetic)

prove the universal statement: pick a number, add 5, multiply by 4, subtract 6, divide by 2, and subtract twice the original number.

x x + 5 4x + 20 4x + 14 2x + 7 7

does 7|42?

yes, 6

is 21 divisible by 3?

yes, 7