Math 310, Proof Portfolio Problems
Problem 1
Consider the puzzle called the Tower of Hanoi (attributed to the French mathematician
Edouard Lucas, 1883). The puzzle consists of n disks of decreasing diameters place don
a pole. There are two other poles. The problem is to move the entire stack of disks to
another pole by moving one disk at a time to any other pole, except that no disk may be
placed on top of a smaller disk. Find a formula for the least number of moves needed to
move a stack of n disks from one pole to another, and prove the formula by induction.
Assigned: Friday 17 January 2014
Problem 2
Choose either Problem 2A or 2B to do.
Problem 2A: Suppose that the function f : R → R satisfies f (xy) = x f (y) + y f (x) for all
real numbers x and y. Prove that f (1) = 0 and that f (un ) = nun−1 f (u) for all positive
integers n and all real numbers u.
Problem 2B: Let f be a function mapping Z into the set of positive real numbers.
Suppose that f (1) = c and that f satisfies f (x − y) = f (x)/ f (y) for all x, y ∈ Z. Find f (n)
for n ≥ 1 an integer and prove that your formula is correct.
Assigned: Wednesday 29 January 2014
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Math 310, Proof Portfolio Problems
Problem 3
Let a, b, c, m be integers, with m ≥ 1. Let d = (a, m). Prove that m divides ab − ac if and
m
only if divides b − c.
d
Assigned: Wednesday 29 January 2014
Problem 4
Choose either Problem 4A or 4B to do.
Problem 4A: Let n be a positive integer (i.e. n ∈ N). Prove that if n ≡ 7 (mod 8), then
n is not the sum of three squares. That is, there do not exist integers a, b, and c such that
n = a2 + b2 + c2 .
Problem 4B: Let a, b, c ∈ Z. Prove that if a2 + b2 + c2 ≡ 0 (mod 5), then at least one of
a, b, and c is divisible by 5.
Assigned: Wednesday 19 February 2014
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Math 310, Proof Portfolio Problems
Problem 5
This problem considers a relation ∼ on the set Q of rational numbers. If a, b ∈ Q, we say
that a ∼ b if and only if a − b is an integer.
(a) Show that ∼ is an equivalence relation on Q. [Note: You should not write your
rational numbers as fractions of integers for this part of the problem. This part is
similar to problem 6 on Exam 1.]
From now on, write a ≡ b (mod 1) if a ∼ b. For notation, we’ll refer to the equivalence
classes of this relation as Q/Z.
(b) Show that every rational number, a, is congruent (mod 1) to a unique rational number ra = bc with 0 ≤ ra < 1. [Note: You may write your rational numbers as fractions
for this part of the problem, and I’d even encourage it (even though I’ve been discouraging the use of fractions everywhere else). Hint: Try thinking about what
division is when working with integers as you prove this.]
We can now define addition and multiplication in Q/Z by working with the representatives
between defined by part (b). Take 0 ≤ ba < 1 and 0 ≤ dc < 1 and define
a c
ac
· = ,
b d bd
a c
ad + bc
+ = the fractional part of
.
b d
bd
and
As an example of addition, we have
1 2 1
+ = in Q/Z,
2 3 6
since in Q, we have 12 + 32 = 76 = 1 16 which has fractional part 16 . The work we did in part
(b) ensures that these operations are well-defined.
(c) Does the distributivity axiom hold for Q/Z with + and · defined this way? That is,
for any a, b, c ∈ Q, is it always true that a · (b + c) = a · b + a · c? If so, provide a
proof. If not, provide a counterexample.
Assigned: Wednesday 19 February 2014
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Math 310, Proof Portfolio Problems
Problem 6
Let R be an integral domain with finitely many elements. Prove that either R = {0} or R
is a field.
Updated: Friday 14 March 2014
Problem 7
Both of these problems involve a special type of subset of a commutative ring, called an
ideal. In a commutative ring, an ideal is a nonempty subset, I ⊆ R which satisfies the
following two properties:
1. We have a + b ∈ I for all a, b ∈ I. (i.e., I is closed under addition)
2. We have r · a ∈ I for all a ∈ I and r ∈ R. (i.e. the product of an element of I with any
element of R is an element of I)
Choose either Problem 7A or 7B to do.
Problem 7A: Let f : R → S be a ring homomorphism. Show that
ker( f ) = {r ∈ R : f (r) = 0}
is an ideal of R.
Problem 7B: Let I be an ideal of a commutative ring R.
(a) Prove that if u ∈ R is a unit and u ∈ I, then I = R.
(b) Prove that 0 ∈ I.
Assigned: Monday 17 March 2014
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Math 310, Proof Portfolio Problems
Problem 8
Let R denote the real numbers, and let Func(R, R) denote the set of all functions
f : R → R.
For two functions f , g ∈ Func(R, R) define f + g via
( f + g)(r) = f (r) + g(r)
for all r ∈ R and define f · g via
( f · g)(r) = f (g(r))
for all r ∈ R. Note that for any f , g ∈ Func(R, R), then f + g, f · g ∈ Func(R, R). Prove
that with these operations the set Func(R, R) is a non-commutative ring.
Updated: Friday 4 April 2014
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