68 terms

Math 300: EXAM 1

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Terms in this set (...)

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.
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