Download Math 3000 Known Theorems/Useful Tautologies Theorem 1.6 If n is

Math 3000
Theorem 1.6
Known Theorems/Useful Tautologies
If n is an odd integer, then n2 is odd.
Theorem 1.7 Suppose the m and b are real numbers with m 6= 0 and that f is the linear function defined by
f (x) = mx + b. If x =
6 y, then f (x) 6= f (y).
Theorem 1.8 Suppose a, b, and c are integers. If a and b are even and c is odd, then the equation ax + by = c
does not have an integral solution for x and y.
Theorem 1.9
If r is a real number such that r2 = 2, then r is irrational.
Theorem 1.11
If a, b, and c are real numbers, then a2 + b2 + c2 ≥ ab + bc + ca.
Theorem 1.12
An integer n is even if and only if n2 is even.
Corollary 1.13
An integer n is even if and only if n3 is even.
Theorem Oddball
Theorem 1.73a
AB is odd if and only if both A and B are odd integers.
If a and b are positive integers such that a2 = b3 and a is even, then 4 divides a.
Tautologies
1. P ∨ Q ←→ ¬P → Q
2. ¬(P → Q) ←→ P ∧ ¬Q
3. ¬(P ∨ Q) ←→ ¬P ∧ ¬Q
4. ¬(P ∧ Q) ←→ ¬P ∨ ¬Q
5. ¬(¬P ) ←→ P
Theorem 4.7 Let n ∈ N and a, b ∈ Z. Then a ≡ b (mod n) if and only if there exists an integer k such that
a = b + kn.
Theorem 6.1 Let n, r ∈ N with 0 ≤ r ≤ n. Then P (n, r) =
n!
(n−r)! .
Theorem 6.2 Let n, r ∈ N with 0 ≤ r ≤ n. Then C(n, r) =
n!
r!(n−r)! .
Theorem 6.8 Let n, r ∈ N with r ≤ n and let x, y ∈ R. Then (x + y)n =
n
X
r=0
C(n, r)xn−r y r .