Download Math 135 Cheat Sheet for Final Exam and more Lecture notes Logic in PDF only on Docsity! Math 135 Cheat Sheet for Final Exam Set Theory Notation empty set ∅ { } subset A ⊆ B ∀x : x ∈ A→ x ∈ B proper subset A ⊂ B A ⊆ B ∧ ∃y ∈ B : y 6∈ A superset A ⊇ B B ⊆ A proper superset A ⊃ B B ⊂ A set equality A = B A ⊆ B ∧B ⊆ A union A ∪B {x | x ∈ A ∨ x ∈ B} intersection A ∩B {x | x ∈ A ∧ x ∈ B} difference A−B {x | x ∈ A ∧ x 6∈ B} = A ∩B symmetric difference A∆B {x | x ∈ A↔ x 6∈ B} complement A {x | x 6∈ A} = U −A Cartesian product A×B {(a, b) | a ∈ A ∧ b ∈ B} power set P(A) {B | B ⊆ A} cardinality |A| # of elements (if finite) Logic proposition statement which is unambiguously true or false predicate proposition which incorporates a variable logical operations and ∧, or ∨, not ¬ universal quantifier for all, written ∀ existential quantifier there exists, written ∃ implication if p then q, written p→ q inverse of p→ q ¬p→ ¬q converse of p→ q q → p contrapositive of p→ q ¬q → ¬p Function f : A → B A function f from A to B associates each element a ∈ A to exactly one element b ∈ B. Notation b = f(a) if b is associated to a one-to-one (or injective) ∀a1, a2 ∈ A, if a1 6= a2 then f(a1) 6= f(a2) onto (or surjective) ∀b ∈ B, ∃a ∈ A such that f(a) = b bijection one-to-one and onto inverse f−1 : B → A {(b, a) | b = f(a)} (if f is a bijection) 1 Recursion tree for T (n) = aT (n/b) + f(n) f(n) f(n/b) f(n/b2) ... . . . f(n/b2) ... . . . f(n/b) f(n/b2) ... . . . f(n/b2) ... −→ f(n) −→ a · f(n/b) −→ a2 · f(n/b2) (logb n levels) Master Theorem Let T (n) = aT (n/b) +O(nk) If a ≥ 1, b is an integer ≥ 1, and k a real number ≥ 0: a < bk =⇒ T (n) = O(nk) a = bk =⇒ T (n) = O(nk log n) a > bk =⇒ T (n) = O(nlogb a) Asymptotic notation f(n) = o(g(n)) ∀c > 0: ∃N > 0: ∀n ≥ N : f(n) < c · g(n) f(n) = O(g(n)) ∃c > 0: ∃N > 0: ∀n ≥ N : f(n)≤ c · g(n) f(n) = Θ(g(n)) f(n) = O(g(n)) and f(n) = Ω(g(n)) f(n) = Ω(g(n)) ∃c > 0: ∃N > 0: ∀n ≥ N : f(n)≥ c · g(n) f(n) = ω(g(n)) ∀c > 0: ∃N > 0: ∀n ≥ N : f(n) > c · g(n) f(n) = O(g(n)) =⇒ f(n) + h(n) = O(g(n) + h(n)) f(n) = O(g(n)) =⇒ f(n) · h(n) = O(g(n) · h(n)) f(n) + g(n) = O(max{f(n), g(n)}) f(n) = O(g(n)) and g(n) = O(h(n)) =⇒ f(n) = O(h(n)) ∞∑ i=0 α = 1 1−α (if α < 1) d∑ i=0 ic = Θ(nc+1) (if c 6= −1) n∑ i=0 ci = Θ(cn) (if c > 1) n∑ i=1 log i = Θ(n log n) Logarithm identities logb(bx) =x blogb x =x logb x= logc x logc b logb(xy) = logb x+ logb y logb(1/x) =− logb x xlogb y = ylogb x logb(xy) = y logb x 2
