CS 161 Final Exam Practice Test - What to study

A\B

"A setminus B" {x|x∈A *and* x ∼⊆ B}

AXB

"The Cartesian product of A *and* B" {(a,b)|a∈A and b∈B}

equipotent

"has the same cardinality"

Cantor's Theorem

#S < #P(S)

claim [0,1] ≈ [3,5]

(5-3)/(1-0) <---- Slope = 2 (a,b) (1,5) y-5 = 2(x-1) ------- f(x) = 2x + 3 claim f is bijective (1-1) let x₁,x₂∈[0,1] and suppose f(x₁) = f(x₂). Then 2x₁ + 3 = 2x₂ + 3. 2x₁ = 2x₂. x₁ = x₂. (onto) let y∈[3,5]. then x = (y-3)/2∈[0,1]. since y∈[3,5] .... f(x) = f((y-3)/2) + 3. = 2((y-3)/2) + 3. = y-3+3

Sets A and B have the same cardinality if

(∃ƒ:A→B)(ƒ is bijective) bijective = 1-1 and Onto

limit

Lim = L ↔ (∀ε>0)(∃δ>0)(|x-c|<δ → |f(x)-L|<ε) x→c

(0,1) ≈ R

f((-π/2),(π/2)) → R defined by f(x) = tan(x) bijective → ((-π/2),(π/2)) ≈ R tan(x - π) + 1. tan(kx). period of tangent = π/k. tan(πx).

"Image of A under ƒ"

f(A) = {b∈B | (∃a∈A)(ƒ(a) = b)}

continuity

f: R → R is continuous at x=c ↔ (∀ε>0)(∃δ>0)(|x-c|<δ → |f(x)-f(c)|<ε

divides

for a,b∈Z: a|b ↔ (∃k∈Z)(b = ak)

complete

has an edge between any two distinct vertices

connected graph

if all verticies are connected

equivalence relation

reflexive: A≈A. symetric: A≈B → B≈A. transitive A≈B and B≈C → A≈C

k-regular graph

when every vertices have the same degree (the same amount of edges coming from it

even

z∈Z is *even* ↔ (∃k∈Z)(z=2k)

odd

z∈Z is *odd* ↔ (∃k∈Z)(z=2k+1) -or- z∈Z is *odd* ↔ (∃k∈Z)(z=2k-1)

"Inverse Image of a set of a under ƒ"

ƒ⁻¹(B₀) = {a∈A | ƒ(a)∈B₀}

f is 1-1

↔ (∀a₁,a₂∈A)(ƒ(a₁)=ƒ(a₂) → a₁ = a₂) "injective"

f is onto

↔ (∀b∈B)(∃a∈A)(ƒ(a) = b) "subjective"