MAT102H5S
Assignment 05 - Winter 2025
Due: Friday, April 4th, 2025 at 11:59pm via Crowdmark.
Late assignments will not be accepted.
Be sure to show and explain all of your work.
1. For each of the following maps, indicate whether the map is injective, surjective, both, or neither. Be sure to prove your claims. For the entirety of this question, 0 is not a natural number.
(a) (2 points) f1 : N × N → N × N, f1(n, m) = (nm, n).
(b) (2 points) f2 : N × N → N, f2(n, m) = n + m − 1.
(c) (2 points) f3 : N × N → [0, ∞), f3(n, m) = m/n.
(d) (4 points) Let F = {a : N → N : a(1), a(2) ∈ N, a(k) = a(k − 1) + a(k − 2), ∀k ≥ 3}. De-fine f4 : N × N → F, f4(n, m) = an,m where
2. For this question, 0 is not a natural number.
Recall that we say that a function f : R → R is strictly increasing if a < b implies that f(a) < f(b).
(a) (2 points) Show that a strictly increasing function is necessarily injective.
(b) (2 points) Show that if f : R → R is an invertible, strictly increasing function, then f
−1 is also strictly increasing.
(c) (2 points) For each n ∈ N, show that f : [0, ∞) → [0, ∞) given by f(x) = x
n
is a strictly increasing function. Note: If you want to use calculus to do this, you must prove all of the necessary calculus results.
For parts (d) and (e), you may freely assume that f : [0, ∞) → [0, ∞), x 7→ x
n
is bijective, and hence invertible. You are free to try the surjectivity proof if you like, but it is beyond the scope of the course.
(d) (2 points) Define the function g : N × [0, ∞) → [0, ∞) as g(n, x) = x
n
. Determine g({3} × [0, 2])?
(e) (2 points) What is g
−1
([0, 1])?