how to write a proof
Prove: [given hypothesis] “assume that…” Proof: show mathematical statements within written lines; Therefore, [calculated conclusion]
how to prove by contrapositive
state/assume not p; begin to prove notq from not p; p and q must be true if not p and not q are false/yield differnt results
how to prove by contradiction
state/assume not p; begin to prove notq from not p; find an untrue statement; p -> q must be true because notp -> notq is false
how to prove by case
recall the rules/definition for that case (ex, even number, n = 2k) and make the problem equivalent(/factorable) to it
prove an existential statement is true: constructuve
find a value of x in D that makes Q(X) true; or generate directions for finding such a value
prove a universal statement is false
form a counterexample for just one x in D
prove an existential statement is true: nonconstructive
show that for some x, Q(x) is true/false via an axiom that would make it true/false;
prove a universal statement is true: exhaustion
if the domain is small, check the value of Q(n) for all n
prove an existential statement is false
prove a universal statement;
prove a universal statement is true: universal generalization
restate as a universal conditional statement (x D, P(x)),
define x: “suppose x in D and P(x).”
state mathematical truths and given assumption about the object, state reasoning in complete sentences
prove a universal statement is true
manipulate mathematical definitions
when to prove by contrapositive
is it easier to prove the negation of the conclusion, or the current hypothesis?
prove no integer is both even and odd
contradiction: there is one n in Z that is even and odd; n = 2k1 and 2k2 + 1; 2k1 = 2k2 + 1 -> k1 - k2 = 1/2; 1/2 != integer, so there is no integer that is both even and odd
|x| =
x if ≥ 0; -x if < 0
theorem
a statement that can be proven to be true
proof
a series of steps following logically from assumptions or previously proven statements
axiom
statements assumed to be true
prove x can be written in two ways as a sum of prime numbers
let x = some composite number. n = (sum1) = (sum 2) where (components) are all prime numbers
prove x^2 > x
x ≥ 1, x^2 = x * x, x * x ≥ 1 * x
prove that the sum of two integers is even
m = 2r, n = 2s; m + n = 2r + 2s = 2(r + s); having two as a factor (like 2k) proves the integers and their sum are even
prove that the square root (/ plain integer) is odd
mathematical definition; plug mathematical definition into n^2 for (2k + 1)^2; factor to express as (something) + 1; since n^2 = 2(2k^2 + 2k) + 1
prove that the difference between an odd and even integer is odd
m = 2r + 1, n = 2s; m - n = (2r + 1) - 2s; rearrange and factor until 2(r - s) + 1; this is the same as 2k + 1, if k = r - s, so m - n is odd by definition
prove that a polynomial is not prime
define prime; factor polynomial; 1 < (r, s) < a
