Miniconference on the Mathematics of Computation MTH 210 Asymptotics II Dr. Anthony Bonato Ryerson University

Big Omega notation assume y = f(x) and y = g(x) are real-valued functions, and lim 𝑥→∞ 𝑔 𝑥 𝑓 𝑥 exists and is finite we write f(x) = Ω(g(x)) also say f(x) is in Ω(g(x)) say: “f(x) is big Omega of g(x)”

Big Theta notation assume y = f(x) and y = g(x) are real-valued functions, and lim 𝑥→∞ 𝑔 𝑥 𝑓 𝑥 exists and is in (0,∞) we write f(x) = Θ(g(x)) also say f(x) is in Θ(g(x)) say: “f(x) is Theta of g(x)”

Key facts f(x) = Ω(g(x)) if and only if g(x) = O(f(x)) f(x) = Θ(g(x)) if and only if f(x) = O(g(x)) and g(x) = O(f(x))

Key facts for all r > s, xr = Ω(xs) anxn+an-1xn-1 + … + a1x +a0 = Θ(xn), where an > 0

Little oh assume y = f(x) and y = g(x) are real-valued functions, and lim 𝑥→∞ 𝑓 𝑥 𝑔 𝑥 = 0 we write f(x) = o(g(x)) also say f(x) is in o(g(x)) say: “f(x) is little oh of g(x)”

Exercises

Miniconference on the Mathematics of Computation MTH 210 Test 2 Review Dr. Anthony Bonato Ryerson University

Notes on Test 2 Test 3 is in-class on Monday, April 22 in KHE117, starting at 12 noon 120 minutes, 7 questions (multiple parts), 40 marks total Material covers all material on: sequences, induction, strong induction, counting, combinations, and asymptotics, up to and including material covered in the April 9 lecture. Need to know: definitions, examples, exercises, assigned problems, quiz material, theorems, key facts Three questions short answer, one question fill in the blank; three questions long answer NOTE: no aids allowed Office hours: Monday, April 15, 12:30 – 1:30 pm

General sequences a1, a2, a3, a4, …, an a1, a2, a3, a4, …, an, … ai are called terms i is the index of ai a1, a2, a3, a4, …, an, … infinite sequence sums and products

Induction want to prove property P(n) holds for all n > 0. verify P(n) in the base case. Say first n is 1. Simply check that P(1) holds. Induction hypothesis: Assume P(n) holds for a fixed n. Induction step: Given the induction hypothesis, show that P(n+1) holds.

Strong Induction same as induction, but assume P(n) is true for ALL values up to a given k. don’t only assume true for k-1

Independent events two events are independent if they do not depend on each other suppose you have k independent events n1 objects from Event 1 n2 objects from Event 2 n3 objects from Event 3 … nk objects from Event k then: number of objects in every event is n1n2 …nk

Pigeonhole property If you have n+1 objects assigned to n properties, then at least two objects have the same properties. NB: Could be “at least n+1”

Combinations n non-negative integer, r ≤ n 𝑛 𝑟 = number of ways to choose r objects from n objects say: “n choose r” call 𝑛 𝑟 a combination or binomial coefficient 𝑛 𝑟 = 𝑛! 𝑛−𝑟 !𝑟!

Pascal’s triangle …

Binomial theorem x, y variables, n non-negative integer

Big Oh notation assume y = f(x) and y = g(x) are real-valued functions, and lim 𝑥→∞ 𝑓 𝑥 𝑔 𝑥 exists and is finite we write f(x) = O(g(x)) also say f(x) is in O(g(x)) say: “f(x) is big Oh of g(x)”

Big Omega notation assume y = f(x) and y = g(x) are real-valued functions, and lim 𝑥→∞ 𝑔 𝑥 𝑓 𝑥 exists and is finite we write f(x) = Ω(g(x)) also say f(x) is in Ω(g(x)) say: “f(x) is big Omega of g(x)”

Big Theta notation assume y = f(x) and y = g(x) are real-valued functions, and lim 𝑥→∞ 𝑔 𝑥 𝑓 𝑥 exists and is in (0,∞) we write f(x) = Θ(g(x)) also say f(x) is in Θ(g(x)) say: “f(x) is Theta of g(x)”

Little oh assume y = f(x) and y = g(x) are real-valued functions, and lim 𝑥→∞ 𝑓 𝑥 𝑔 𝑥 = 0 we write f(x) = o(g(x)) also say f(x) is in o(g(x)) say: “f(x) is little oh of g(x)”

Key facts for all r < s, xr = O(xs) anxn+an-1xn-1 + … + a1x +a0 = O(xn), where an > 0 for all r > s, xr = Ω(xs) anxn+an-1xn-1 + … + a1x +a0 = Θ(xn), where an > 0