Proof Flashcards
(12 cards)
Proof letters
N = Natural Numbers
Z = Integers
Q = Rational numbers /. Quotient
R = Real numbers
Implies other phrases
P => Q | P is sufficient for Q | P, only if Q
Q => P | P is necessary of Q | P, if Q
P <=> Q | P is necessary and sufficient for Q | P, iff Q| P, if and only if Q
“In the range/interval”
Proof by exhaustion
X = 1, Workings, Conclusion
X = 2,Workings, Conclusion
ect
“Counter example”
2 only even prime
0, 1, -1 behave differently
sqrt being rational or irrational
Contradiction
1) “Assume opposite is true”
2) Prove assumption
3) Highlight contradiction
4) Conclusion, “This is a contradiction of the assumption: X, therefore original statement
Prove sqrt(2) is irrational
1: Sqrt(2) = a/b
2: 2 = a^2/b^2
3: 2(b^2) = a^2
4: Therefore a is even
5: 2k = a
6: Therefore b is even
7: a and b share a factor of 2
Prove inf primes
1) Assume finite primves
2) Primes are p1, p2, p3, pn(largest prime)
3) Multiple together and add 1
4) This number is bigger than primes so cant be prime
5) So it must be composite (not prime)
6) Divide by each prime
7) Can always be written as X+1/p1
8) Not composite
“Two digit number”
10a + b where a 1 < 9 and b 0 < 9
“Rational number”
a/b where a nad b are integers and b != 0
“Any 2 integers”
n, m
“Two”
n, m
“At least one”
Neither
