MATH 582 HOMEWORK 1
WEEK 2
Winter, 2009
Due January 23
Exercise 1. Show that the following identity need not hold when f :
A B is not injective.
f [X Y ] = f [X] f [Y ]
X, Y A.
Proof. Let A = B = cfw_0, 1 and dene f : A B by f (0) = f (1) = 0.
Let X = cfw
MATH 582 HOMEWORK 2
WEEK 3
Winter, 2009
Due January 23
Exercise 1. For all sets A and B,
(a)
(b)
A B P(A) P(B).
A B A2 B 2
Proof. (a). Suppose : A B is a bijection which witnesses A B.
Then the map X [X] denes a bijection from P(A) to P(B).
The map is in
MATH 582 HOMEWORK 3
WEEK 5
Winter, 2009
Due February 20
Exercise 1. Let (A, <) and (B, ) be ordered sets with A B = .
Dene on A B as follows: for any x, y A B let x y i either
(i) x, y A and x < y, or
(ii) x, y B and x < y, or
(iii) x A and y B
(The relat
MATH 582 HOMEWORK 3
WEEK 6
Winter, 2009
Due February 20
Exercise 1. Prove the following:
Suppose (N1 , 01 , S1 ) and (N2 , 02 , S2 ) are two systems of natural numbers, where +1 , +2 are their respective canonical operations of addition.
Then the canonica
MATH 582 HOMEWORK 1
WEEK 1
Winter, 2009
Due January 23
Problem 1.
Exercise. Let A, B and C be sets. Show the following.
(a) A C B C A B C.
(b) C A C B C A B.
Proof. (a). Suppose A C and B C. Fix any x with x A B.
Then, either x A or x B. If x A then x C b
MATH 582 HOMEWORK 2
WEEK 3
Winter, 2009
Due February 6
1. The language of sets consists of a single binary relation cfw_ . An
interpretation (or a model ) of the language of sets consists of a pair
(D, E), where D is the domain of discourse and E is a bin
MATH 582 HOMEWORK 4
WEEK 8
Winter, 2009
Due March 13
Exercise 1. A set z is a transitive set if and only if
z z.
Proof. (). Suppose z is a transitive set. If x z, then x y for
some y z. But y z (transitive set), so x z. Since x was arbitrary,
z z.
(). Sup
MATH 582 HOMEWORK
WEEK 14
Winter, 2009
Due April 20
Introduction
This is a full complement of problems. Let your time and interests
dictate how much you choose to do. The most important problems (and
most interesting I hope) are problems 6-9 on equivalent
MATH 582 HOMEWORK 5
WEEK 9
Winter, 2009
Due March 27
Exercise 1. Prove the distributive law : for all , , ,
( + ) = + .
Proof. The proof is by transnite induction on . When = 0 both
sides of the equality are 0. Suppose = + 1, and that (i.h.)
( + ) = + .
MATH 582 HOMEWORK 6
WEEK 10
Winter, 2009
Due April 10
Exercise 1. Counting arguments.
(a) A sequence an n < of natural numbers is eventually constant if
there is a k such that sk = sn for all n k. Show that the set
of eventually constant sequences of natu
MATH 582 HOMEWORK 4
WEEK 7
Winter, 2009
Due March 13
Exercise 1. Recall the denition of multiplication from HW3 on Week
6 (Exercise 2). Prove that multiplication respects order: for every
m, n, k
k>0
m<n mk <nk .
Hint. I have not proven all the intermedi
MATH 582 HOMEWORK 7
WEEK 12
Winter, 2009
Due April 20
1. Axiom of Choice and Equivalents
You may use the Axiom of Choice (AC) or one of its equivalents
from Lecture 28.
Exercise 1. Prove the following statement is equivalent to the Axiom
of Choice:
Any su