# All pages (Proof namespace)

From CS2800 wiki

- Proof:A/R partitions A
- Proof:A = (A ∩ B) ∪ (A ∖ B)
- Proof:A ∩ (B 1 ∪ ... ∪ B n) ⊆ (A ∩ B 1) ∪ ... ∪ (A ∩ B n)
- Proof:A ∩ (B∪C) ⊆ (A∩B) ∪ (A∩C)
- Proof:A ∩ B ⊆ A
- Proof:A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- Proof:A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (example of poor style)
- Proof:A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) bogus proof (by example)
- Proof:Base b representation
- Proof:Bayes' rule
- Proof:Bézout coefficients exist
- Proof:Cantor-Schroeder-Bernstein theorem
- Proof:Cardinality of evens
- Proof:Cardinality of the integers
- Proof:Euclidean division algorithm
- Proof:Every natural number has a prime factorization
- Proof:Every natural number has a prime factorization (strengthened induction hypothesis)
- Proof:Every natural number has a prime factorization (weak induction; doesn't work)
- Proof:Functions with left inverses are injective
- Proof:Gcd(a,b) is a common divisor of a and b
- Proof:Gcd(a,b) is greater than all other common divisors of a and b
- Proof:If A ⊆ B and B is countable then A is countable
- Proof:If p is prime, then φ(p) = p - 1
- Proof:Injections have left inverses
- Proof:Law of total probability
- Proof:Limit of x at 0 is 0
- Proof:Limit of x at 0 is not 1
- Proof:Pr(E) ≤ 1
- Proof:Pr(S ∖ E) = 1 - Pr(E)
- Proof:Random variables satisfy distributivity
- Proof:Strong induction is equivalent to weak induction
- Proof:Surjections have right inverses
- Proof:The base b representation is unique
- Proof:The power set of the naturals is uncountable
- Proof:The quotient and remainder are unique
- Proof:The set of reals is uncountable
- Proof:There is a unique function with an empty domain
- Proof:Two definitions of independence are equivalent
- Proof:Union computation
- Proof:Weak induction principle starting at k is equivalent to weak induction
- Proof:Weak induction principle with n-1 is equivalent to weak induction
- Proof:Σi = n(n+1)/2
- Proof:∅ ∉ ∅
- Proof:∅ ⊆ A
- Proof:≤ and ≥ are related
- Proof:≤ is reflexive
- Proof:≤ is transitive
- Proof:│ℕ ∪ -1│ = │ℕ│
- Proof:│ℕ ⨯ ℕ│ = │ℕ│
- Proof:⟦a⟧=⟦b⟧ if and only if aRb
- Proof:⟦a⟧ is a unit mod m if and only if gcd(a,m) = 1