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

2. 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)”

3. 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)”

4. 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))

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

6. 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)”

7. Exercises

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

9. 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

10. 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

11. 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.

12. 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

13. 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

14. 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”

15. 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!

16. Pascal’s triangle …

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

18. 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)”

19. 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)”

20. 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)”

21. 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)”

22. 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