ch 3,4,5 Flashcards
trivial proof
if it can be shown that Q is always true regardless of the truth value of P
vacuous proof
if it can be shown that P is false for all x in S regardless of the truth value of Q
direct proof
assume that P is true for an arbitrary element x in S and show that Q must be true for this element x
contrapositive
the contrapositive of the implication P implies Q is (~Q) implies (~P)
proof by cases
verifying the truth statement for each property x may have
case
a case for each property that x may possess or for each subset to which x may belong
subcases
when a case is further divided into other cases
parity
the property of an integer being either even or odd
theorem
a mathematical statement that has been proven to be true
axiom
a statement accepted as true without proof
without loss of generality (WLOG)
indicates that the proofs of the two situations are similar, so the proof of only one is needed
divides/divisibility
a divides b if there is an integer c such that b = ac. written a | b
proof by contradiction
assuming the negation of a statement and deriving a contradiction thus proving the original statement to be true
lemma
a math statement proven to support the proof of a theorem
corollary
a math statement proven as a consequence of a theorem
conjecture
a proposed math statement that has not been proven
