21-127 Concepts

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smartin014  on April 26, 2011

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21-127 Concepts

 Bounded SetFor all elements s in the set, there exists a real value M such that |s| < M
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Bounded Set For all elements s in the set, there exists a real value M such that |s| < M
Increasing Function f(x) < f(x') whenever x < x'
Decreasing Function f(x) > f(x') when x < x'
Nonincreasing Function f(x) ≥ f(x') whenever x < x'
Nondecreasing function f(x) ≤ f(x') whenever x < x'
Set Equality Prove S⊆T, then prove T⊆S.
Do this by considering the first set, then deriving the second.
AGM Inequality If x,y ∈ R then 2xy ≤ x^2 + y^2
and xy ≤ ((x+y)/2)^2
If they are both non-negative, (xy)^.5 ≤ (x+y)/2
Triangle Inequality |x + y| ≤ |x| + |y|
g composed with f f: A->B, g: B->C
g o f => h: A->C, h(x) = g(f(x)) for x ∈ A.
Composition of two surjections surjection
Composition of two injections injection
Composition of two bijections bijection
Surjective function f: A-> B,
∀y ∈ B, ∃x∈A such that f(x) = y
Injective Function f: A-> B.
∀x,y ∈ A, f(x) = f(y) => x = y
Well-ordered Set Has a least element
Direct Proof Assume P, derive Q
Proof by Contrapositive Assume not Q, derive not P
Proof by Contradiction Show that P and not Q can never be true
Proof by Induction Base case: P(1) is true
Inductive Hypothesis: Assume true for some n.
Show P(n) => P(n+1)
(g o f)^(-1) f^(-1) o g^(-1)
f has an inverse is logically equivalent to f is a bijection, and implies that the inverse is unique.
Countability A is countable iff A is finite or there exists a bijection between A and a set known to be countable.
Cardinality If there exists a bijection between A and B, |A| = |B|.
Known countable sets The natural numbers, integers, rational numbers are countable.
The real numbers and the irrational numbers are uncountable.
PowerSet(A) PowerSet(A) has the cardinality 2ⁿ if A has the cardinality n.
Diagonalization Proof technique to disprove surjectivity. Ensure every ƒ from A to B will miss an element of B.
Snaking Proof technique to show countability. Show that every element will be hit in a finite number of steps.
Monotone An increasing or decreasing function. (A strictly monotone function is strictly increasing or decreasing.)
Canter-Bernstein-Schroder Theorem If ∃ injections from A to B and from B to A, then A and B are bijections.
Injectivity and Cardinality ƒ: A -> B injective => |A|≤|B|
Surjectivity and Cardinality ƒ: A -> B surjective => |A|≥|B|
Partition 1. The intersection of any two distinct sets is the empty set.
2. The union of all sets is the universe.
Permutation A permutation of S is the bijection from S to S. An n element set has n! permutations.
Permutations with repetition For n element, k of which are distinct: n!/(n-k)!
n choose k n!/(k!(n-k)!)
Used for cases where order/distinct elements don't matter.
Chairperson Identity k(n choose k) = n( (n-1) choose (k-1))
Summation Identity sum from{i = 0} to{n} (i choose k) = (n+1) choose (k+1)
(n choose k) is also n choose (n-k)
Pascal's Identity (n choose k) + (n choose (k-1)) = ((n+1) choose k)
Pascal's Triangle 2ⁿ = sum of elements in each row, from the 0th row.
Symmetric.
The nth row has elements from (n choose 0) to (n choose n).
Stars and Bars (n + k - 1) choose (k - 1)
given n objects being distributed in k groups with empty groups possible.
Binomial Theorem (x+y)ⁿ = sum from{k = 0} to{n} (n choose k) x^(n-k) y^k
Notations for permutations Two line, cycle.
Do permutations commute? Only disjoint cycles can commute.
Identity permutations (1), usually not written
Transposition Two element permutation
Permutation as combination of transpositions (x₁ x₂ ... x₋) = (x₁ x₋) ... (x₁ x₃) (x₁ x₂)
If a|b and b|c, then a|c.
If c|a and c|b, then c| (ma+nb)
PI(x) Number of primes ≤ x
Euclidean Algorithm 1. (a,b) = (b,r) if r = a - bq
2. (a,0) = a
Fundamental Theorem of Arithmetic All natural numbers bigger than 1 can be written uniquely as products of powers of primes.
Linear Diophantine Equation:
ax + by = c, (a,b) = d
Only has solution if d|c.
Solutions: if x₀, y₀ are solutions, then all x = x₀ + (b/d)n and y = y₀ + (a/d)n are solutions too.
R is a binary relation on A and B iff R ⊆ A x B
Notation: (a,b) ∈ R R(a,b) = aRb
R is an equivalence relation [R(a,b) is (a~b)] iff it fulfills reflexivity (xRx), symmetry (xRy => yRx), and transitivity (xRy, yRz => xRz)
a ≡ b (mod m) iff m | (a-b)
Z_m Set of congruence classes mod m
a ≡ b (mod m) => a + c ≡ b + c (mod m), and
ac ≡ bc (mod m)
Division in modular arithmetic (c,m) = d, ac ≡ bc (mod m) =>
a ≡ b (mod (m/d))
a ≡ b and c ≡ d (mod m) => a + c ≡ b + d (mod m), and
ac ≡ bd (mod m)
Linear Congruences ax ≡ b (mod m), (a,m) = d.
d|b <==> d incongruent solutions
x = x₀ + (m/d) t, 0≤t≤(d-1)
Chinese remainder theorem Given equivalences mod pairwise prime numbers, there exist numbers fulfilling all conditions mod the product of all the pairwise prime numbers.
Modular Inverses ax ≡ 1 (mod m) given (a,m) = 1
Wilson's Theorem If p prime then (p-1)! ≡ -1 (mod p)
Fermat's Little Theorem p prime, a integer, a not divisible by p =>
a^(p-1) ≡ 1 (mod p)
FLT corollaries 1. a^(p-2) is the inverse of a (mod p)
2. ax ≡ b (mod p) => x ≡ ba^(p-2) (mod p)
P(AUB) P(A) + P(B) - P(A∩B)
Bertrand's Ballot Problem: 'a' gets more votes than 'b', what is the probability 'a' never trails during the election? (a-b+1) choose (a+1)
P(A|B) P(A∩B) / P(B)
Mutually exclusive events P(A∩B) = 0
Bayes' Theorem P(A_i|B) = P(A_i∩B) / P(B),
P(B) = P(B|A₁) P(A₁) + P(B|A₂) P(A₂) + ...
Independence Proofs 1. P(A|B) = P(A)
2. P(B|A) = P(B)
3. P(A∩B) = P(A)P(B)
Expected Value Sum over all possible events of P(event)*Value(event)
Multinomial Coefficient (n choose k₁, k₂, k₃ ...) = n! / (k₁! k₂! k₃!...)

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