Math 2200 Homework 3
Due Date: September 25
Please write neatly and leave enough space so I can write comments! Remember that there are solutions
to all the odd-numbered exercises in the textbook. I will mainly be looking at your solutions to the evennumbered exercises and the additional problem, so make sure you attempt most of those (especially the extra
problem!). For the proofs in the additional problem, please state the theorem being proved, write “proof”,
and then give your argument using complete English sentences!
Section 2.2 (p. 136-7)
# 29, # 30, # 51.
Section 2.3 (p. 152-3)
# 1, # 2, # 15, # 22, # 23, # 36.
Section 2.5 (p. 176)
# 1, # 6.
Additional Problem: Proofs with functions
! Prove the following two theorems:
Theorem. If A
Theorem. If A
f
f
! B and B
! B and B
g
g
! C are injective functions, then A
! C are surjective functions, then A
g◦f
g◦f
! C is injective.
! C is surjective.
Hint: use direct proofs and the definitions of injective and surjective functions in terms of elements:
Definition. A function A
f
A function A
f
! B is injective if for any elements a, a" ∈ A, if f (a) = f (a" ) then a = a" .
! B is surjective if for each b ∈ B there is some a ∈ A such that f (a) = b.
The two theorems above immediately give the following corollary (a corollary is a theorem whose proof
easily follows from another theorem):
Corollary. If A
lently, if A
f
f
! B and B
! B and B
g
g
! C are bijective functions, then A
! C are invertible functions, then A
g◦f
g◦f
! C is bijective. Equiva-
! C is invertible.
! In the context of the corollary, what is the inverse of g ◦ f in terms of the inverses f −1 and g −1 of f and
g? Optional: state your result as a theorem and prove it!
1