68 terms

statement

aka proposition: declarative sentence that is either true or false, but not both

conditional statement

a statement in the form "If p, then q" where P and Q are sentences (P is "hypothesis"; Q is "conclusion")

prime number

a positive integer p > 1 whose only positive divisors are 1 and p

If n is a positive integer, then n^2 - n + 41 is a prime number.

False: counterexample n = 41. In this case, etc...

closure properties of number systems (order)

N (natural numbers) is in Z (integers) is in Q (rational numbers) is in R (real numbers)

closure properties for addition, subtraction, and multiplication hold for...

integers, rational numbers, and real numbers

direct proof (of a proposition)

shows that the proposition follows logically from definitions and previously-known propositions

definition

an agreement on the meaning of a certain word or phrase

even

integer a is even provided there exists an integer n such that a = 2n

odd

integer a is odd provided there exists an integer n such that a = 2n + 1

theorem

statement that is important

If x and y are odd integers then xy is odd.

direct proof

If m is a real number and m, m+1, m+2 are the lengths of the three sides of a right triangle, then m = 3.

direct proof

logical operators

way of combining several statements to make new one

compound statement

result of combining several statements to make new one using logical operators

types of logical operators

conjunctions (P and Q), disjunctions (P or Q), negation (not P), implication (if P, then Q)

axiom

mathematical statement without proof (like for us, closure properties)

conjecture

plausible, not proven yet

lemma

true math statement proven mainly to help in proof of some theorem

corrolary

consequence of a proven proposition

different ways to say "if P, then Q"

Q if P; P implies Q; if P is true, Q is true; P is sufficient for Q; Q is necessary for P; P only if Q

tautology

compound statement that is true for all possible combinations of truth values of its components

contradiction

compound statement that is false for all possible combinations of truth values of its components

logically equivalent

two compound statements are logically equivalent provided they have the same truth value for all possible combinations of truth values of its components

converse

of P>Q is Q>P; not logically equivalent

contrapsoitive

of P > Q is not Q > not P; logically equivalent

DeMorgan's Law

not (P and Q ) === P or (P or Q); not (P or Q) === P and Q

list of important logical equivalencies

DeMorgan's Law, conditional statement, biconditionals, double negation, distributive laws, conditionals with disjunctions

set

well-defined collection of objects

roster method

listing elements of a set inside braces

set symbols

E is "is element of" ; C- is "is a subset of" (A is a subset of B if every element of A is in B); = when have same elements even if not in same order or there are multiples

empty set

set that contains no elements

open sentence

sentence with a variable such that it is not a statement; also called predicate or prepositional function

can turn an open sentence into a sentence by...

choosing x from a given set U called the "universal set"

set builder notation

alternative to roster method; where sets and open sentences interact; requires universal set U and a predicate P(X); set of all objects in universal set such that P(X) is true

universal quantifier

the phrase "for every"; denoted by upside down A; usees comma

existential quantifier

the phrase denoted by "there exists...such that"; denoted by backwards E

perfect square

a natural number n is considered a perfect square if there exists an integer k such that n = k^2

divides

a nonzero integer m divides an integer n provided there exists an integer q such that n = mq. In this case, we write m | n and m is the divisor while n is the multiple

truth set of an open sentence

collection of objects in universal set that make predicate true aka all the values the variable could take to make the predicate true

dividing theorem

Let a, b, c be integers with a nonzero and b nonzero. If a|b and b|c then a|c. direct proof

congruence

concept used to describe cycles in the world of integers

if it is a list of numbers like 2, 9, 16, etc...

any 2 numbers are congruent modulo 7

Wilson's Theorem

Let n > 1 be a natural number; then n is prime if and only if (n-1)! = -1(mod n)

For every integer a, b, if a = 5(mod 8) and b = 5(mod 8) then a+b = 2(mod 8)

direct proof...by definition of congruence...divides and hence mod

odd theory

Let n be a natural number. If n^2 is odd, then n is odd; proof by contrapositive; We assume n is even. We want to prove that n^2 is even.

Let a, b be real numbers. If a is nonzero and b is nonzero, then ab is nonzero.

proof by contrapositive; if b is zero then we are done so we take b as nonzero and divide by b

Let x be a real number. We want to prove x equals 2 if and only if x^3-2x^2+x=2.

proof of biconditional ("We will prove this biconditional by proving the 2 following conditional statements...")

Let n be an integer. The following hold: n is even (if and only if) n^2 is even

The first one...We will prove this biconiditonal by proving the two following conditional statements. The first one uses a direct proof; the second one uses proof by contrapositive.

constructive and nonconstructive proofs

both used to prove existence theorems; consstructive shows that there exists an x in the universal set such that P(X) and you directly name/describe this object; nonconstructive shows that there exists an x in the universal set such that P(X) without naming it explicitly

If a, b, and c are real numbers and a is nonzero, ax + b = c has exactly 1 solution.

constructive proof...solve for x= (c-b)/a and then prove that if you plug in ax + b = c it works which means that ax + b = c has a UNIQUE SOLUTION and it is (c-b)/a

What can be solved using nonconstructive proof?

Intermediate Value Theorem: For a continuous function f on the interval [a, b] for a real number q in between f(a) and f(b), there exists a c in [a, b] such that f(c) = q. For example, for f(x) = x^3 -x+1 rom [-2, 0], because f(-2) = -5 and f(0) = 1,

arguing by contradiction means...

arguing that negation of x leads to a contradiction, and hence that x is true; could be contradiction of an assumption or contradiction of what we already know to be true

arguing by contraposiitve means...

using direct proof to prove contrapositive

Let xy be real numbers. If x is not equal to y, x and y are greater than 0, then x/y + y/x is greater than 2.

proof by contradiction; turns into "x is not equal to y, x and y are greater than 0, AND x/y + y/x is less than or equal to 2."

Let x and y be integers. If x and y are odd, then there does not exist an integer z such that x^2 + y^2 = z^2.

proof by contradiction; "We assume that x and y are integers, x and y are odd and there exists an integer z such that x^2 + y^2 = z^2"; use the fact that an odd^2 is odd and that two odds added together are even and that 4 cannot divide a negative number

rational number

a real number x is a rational number provided there eixsts an m and a nonzero n such that x = m/n

irrational numbers are not...

closed under anything

What is true about product of rational or irrational numbers? How prove?

Let x, y be real numbers. If x is rational and nonzero and y is irrational, then xy is irrational. proof by contradiction

The real number square root of 2 is irrational.

proof by contradiction

For all integers a and b, if ab is even then a is even or b is even.

contrapositive

For each integer a, a = 2 mod 8 if and only if a^2 + 4a = 4mod8.

biconditional; first of which is a direct proof

For all integers a and m, if a and me are the lengts fo the sides of a right ttriangle and m+1 is the length of the hypotenuse then a is an odd integer.

direct proof

If r is a real number such that r^3 is 2, then r is an irrational number.

contradiction

For each real number x, x(1-x) <- 1/4.

contradiction

There are no integers a and b such that b^2 = 4a + 2.

contradiction

contradiction examples

There are no integers a and b such that b^2 = 4a + 2.

For each real number x, x(1-x) <- 1/4.

If r is a real number such that r^3 is 2, then r is an irrational number.

The real number square root of 2 is irrational.

For each real number x, x(1-x) <- 1/4.

If r is a real number such that r^3 is 2, then r is an irrational number.

The real number square root of 2 is irrational.

We assume tht a and b are integers, a = 7 mod 8 and b = 3 mod 8. We want to prove that ab = 5mod8.

direct proof