discrete math test 4

d|n explain this notation

"d divides n". symbolically if n and d are integers with d ≠ 0 d|n ↔ ∃ an integer k such that n =dk

how disprove an existential?

must disprove the negation which is a universal (so direct proof)

apply the quotient remainder theorem to n with d = 4. what does this take place in?

n = dq + r is the general form. remainder must be 0≤r<d n = 4q or n = 4q + 1 or n = 4q + 2 or n = 4q + 3 proof by division into cases

find n mod d using div

n mod d = n - d*(n div d)

fermat numbers

2^2^n + 1

list all the divisors of 1

1 and -1

3 steps for proof by contraposition

1. express statement to be proved in the form ∀x in D,if P(x) then Q(x) 2. Rewrite this statement in the contrapositive form ∀x in D,if Q(x) is false then P(x) is false 3. Prove the contrapositive by a direct proof a) suppose x is any element of D such that Q(x) is false b) show that P(x) is false

3 steps for proof by contradiction

1. suppose the statement to be proved is false. That is, suppose that the negation of the statement is true. 2. show that this supposition leads logically to a contradiction. 3. conclude that the statement to be proved is true.

mersenne primes

2^p -1

definition of rational

a real number r is rational if, and only if, it can be expressed as a quotient of two integers with a nonzero denominator. symbolically, r is rational ⇔ ∃ integers a and b such that r a/b and b ≠ 0

if a and b are positive integers and a divides b, then what?

a ≤ b

alternate definition of primes using divisor

an integer n > 1 is prime if and only if its only positive integer divisors are 1 and itself.

definition of composite

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 symbolically n is composite ⇔∃positive integers r and s such that n=rs and 1<r<n and 1<s<n

definition of an even integer

an integer n is even if, and only if, n equals twice some integer. Symbolically n is even iff ∃ an integer k such that n=2k

definition of an odd integer

an integer n is odd if, and only if, n equals twice some integer plus 1. Symbolically n is odd iff ∃ an integer k such that n=2k + 1

definition of prime

an integer n is prime if, and only if, n>1 and for all positive integers r and s, if n = rs, then either r or s equals n. symbolically, nis prime ⇔∀positive integers r and s,if n=rs then either r=1 and s=n or r=n and s=1.

what is standard factored form and why does it work?

any integer n > 1 can be written as prime factors written in ascending order from left to write because of the unique factorization of integers theorem.

definition of irrational

any real number that is not rational

why is the number one not allowed to be prime?

because the theorem of unique factorization would not hold true. for example 6=2*3=1*2*3

prove a universal statement?

direct proof

product of two even integers

even

step for method of direct proof

express the statement to proved in the form ∀x∈D,ifP(x)then Q(x) -start the proof by supposing x is a particular but arbitrarily chosen element of D for which P(x) is true. In other words, Suppose x∈DandP(x) -show that the conclusion Q(x) is true by using definitions, etc

prove an existential statement

find an example that is true

disprove a universal?

find one counterexample.

zero product property

if any of two real numbers is zero, then their product is also not zero.

definition of divisibility

if n and d are integers and n ≠ 0 then n is divisible by d if, and only if, n equals d times some integer

the sums and products of integers are

integers

if a and b are ints. if a|b and b|a does a=b?

no because of a counter example like a = -2 and b = 2 they both divide each other but -2 not equal 2

can negative numbers be prime?

no because this would make unique factorization messed up like 4 = 2*2 and 4 = -2*-2 so negative numbers not considered prime

even int minus odd int

odd

odd int minus even int

odd

starting steps for proving n^2 can be written in the form 8m+1

quotient remainder theorem try setting d as 4 and you get n = 4q or n = 4q + 1 or n = 4q + 2 or n = 4q + 3 pick n = 4q + 1 and n = 4q + 3 because they are odd. do case 1 as n^2 = (4q + 1)^2 and simplify by algebra to 8(2q^2 + q) + 1 which is just 8m + 1 since the middle is just integer sum. do same with n = 4q + 3 for case 2.

basic concept of the unique factorization of integers theorem

says that any integer greater than 1 either is prime or can be written as a product of prime numbers in a way that is unique except, perhaps, for the order in which the primes are written. like 72=2·2·2·3·3=2·3·3·2·2=3·2·2·3·2

prove the sum of any two even ints is even

suppose m and n are any even integers. by definition of even m = 2r and n = 2s for some ints r and s. then. m + n = 2r + 2s =2(r+s) we know r+s is integer since sum of integer is integer 2(integer) is even by definition of even integers

the unique factorization of integers is also known as

the fundamental theorem of arithmetic

n div d

the integer quotient obtained when n is divided by d

n mod d

the nonnegative integer remainder obtained when n is divided by d

the sum, product and difference of any two even integers are even

true

the sum, diff, and product of two rational numbers is rational? what is this called

true. we say that rational numbers are closed under addition subtraction and multiplication.

definition of quotient remainder theorem

when any integer n is divided by any positive integer d, the result is a quotient q and a nonnegative remainder r that is smaller than d. symbolically, n=dq+r and 0≤r<d

product of any two odd integers is odd?

yes

is the double of a rational number rational?

yes because 2r = r + r and by the sum of rational numbers above, that is rational

is ever integer a rational number

yes because integer = n /1

is the sum of rational numbers rational? how proof

yes. r and s. r = a/b s = c/d r + s = common denom of the two and then say integer/integer so rational.

all numbers greater than 1 either prime or composite?

yes. because all possibilities for n = rs is covered by r = 1 and s = n combinations and by the 1<r<n and 1<s<n conditions.

is division transitive? explain

yes. if one number divides a second and the second number divides a third, then the first number divides the third.

are rationals closed under division

yes. only nonzero rationals though.