Homework 6 - Due Tuesday October 28
MTH230 Fall 2014.
(1) State and describe at least three dierent familiar equivalence relations on the set
of triangles in the plane.
(2) Recall that Q = cfw_ a : a, b Z and b = 0 where we use = for the equivalence
b
rel
Homework 1 - Due Tuesday September 2
MTH230 Fall 2014.
(1) What are the possible sums of angles of triangles on a sphere?
(What do we even mean by triangle here?)
(2) Theorem: If A, B, C are points on a circle and AC is a diameter, then ABC is a right ang
Homework 10 - NOT Due
MTH230 Fall 2014.
(1) Recall that for a positive integer m, the number of positive integers less than or equal
to m that are coprime to m is (m). Prove the following. (Try counting the number
of positive integers less than m divisibl
Homework 2 - Due Tuesday September 9
MTH230 Fall 2014.
(1) Prove that 3 is not a ratio of two integers, in two steps:
(a) First prove the Lemma: Give an integer n, if n2 is a multiple of 3, then so is n.
[Hint: Prove the contrapositive. Note that there ar
Homework 4 - Due Tuesday September 30
MTH230 Fall 2014.
(1) Divisibility.
Prove the following
(a) If ac|bc and c = 0, then a|b.
(b) For all a, b Z, if a|b and b = 0 then |a| |b|.
(This is Prop 2.11(iv) in the book. Try working it out yourself before readi
Homework 7 - Due Tuesday November 4
MTH230 Fall 2014.
(1) Modular Arithmetic. Fix an integer n 2. Lets use the notation a n b to mean
that n|(a b).
(a) Show that if gcd(c, n) = 1 then there exists a number x Z such that cx n 1.
(b) Use the above to prove
Denition of the Integers
MTH230 Fall 2014.
(based o notes of Armstrong)
Denition 1
Z := cfw_. . . , 2, 1, 0, 1, 2, . . .
Denition 2
Let Z be a set equipped with
an equivalence relation = dened by
(reexive) a Z, a = a
(symmetric) a, b Z, a = b = b = a
Homework 8 - Not Due
MTH230 Fall 2014.
(1) Let a, q be real numbers, with q = 1. Use induction to prove that for all integers
n 1 we have
a(1 q n )
a + aq + aq 2 + + aq n1 =
.
1q
Can you nd a proof that doesnt require induction?
(Also: What happens if you
Homework 5 - Due Thursday October 9
MTH230 Fall 2014.
(1) Recall that two integers are coprime if their gcd is 1.
Prove that ab is coprime to c if and only if a and b are both coprime to c.
(2) For all integers a, b Z with b = 0, we dene an abstract symbo
Homework 3 - Due Tuesday September 23
MTH230 Fall 2014.
(1) Counting Fingers:
(Assuming you yourself actually have 5 ngers on each hand.)
(a) How many functions are there from the ngers on your right hand to the ngers
on your left hand?
(b) How many bijec
Math 230 E
Homework 6
Fall 2013
Drew Armstrong
Problem 1 (Binomial Theorem). We proved in class that for all n
(1 + x)n =
n
X
k=0
0 we have
n!
xk .
k ! (n k )!
Use this to prove that for all integers a, b 2 Z we have
( a + b) n =
n
X
k=0
n!
an
k ! (n k )!
Math 230 E
Homework 5
Fall 2013
Drew Armstrong
Problem 1. Use induction to prove that for all integers n 1 we have
13 + 23 + 33 + + n3 = (1 + 2 + + n)2 .
This result appears in the Aryabhatiya of Aryabhata (499 CE, when he was 23 years old).
[Hint: You ma
Math 230 E
Homework 4
Fall 2013
Drew Armstrong
Problem 1. Prove that for all integers a, b Z we have
(ab = 0)
=
(a = 0 or b = 0).
You may assume the following axioms: (1) For all x, y, z Z, if x < y and z > 0 then xz < yz .
(2) For all x, y, z Z, if x < y
Math 230 E
Exam 1 Fri Oct 4
Fall 2013
Drew Armstrong
There are 4 problems, worth 5 points each. This is a closed book test. Anyone caught
cheating will receive a score of zero.
Problem 1. Let a, b, q, r Z be integers such that a = qb + r, and consider any
Math 230 E
Exam 2 Fri Nov 8
Fall 2013
Drew Armstrong
There are 4 problems, worth 5 points each. This is a closed book test. Anyone caught
cheating will receive a score of zero.
Problem 1.
(a) Use the Euclidean Algorithm to show that gcd(41, 12) = 1.
gcd(4
Math 230 E
Homework 1 Solutions
Fall 2013
Drew Armstrong
Problem 1. Proposition I.5 in Euclid has acquired the name pons asinorum, which translates
as bridge of asses or bridge of fools. Apparently, many students never got past this proposition. (I would
Math 230 E
Homework 2
Fall 2013
Drew Armstrong
Problem 1. Practice with the axioms of Z. For the following exercises I want you to
give Euclidean style proofs using the axioms of Z from the handout. That is, dont assume
anything and justify every tiny lit
Math 230 E
Homework 3 Solutions
Fall 2013
Drew Armstrong
Problem 1. Let X and Y be nite sets.
(a) If there exists a surjective function f : X Y , prove that |X | |Y |.
(b) If there exists an injective function g : X Y , prove that |X | |Y |.
(c) If there
Homework 9 - Due Tuesday Dec 2
MTH230 Fall 2014.
(1) Use induction to prove that for n 2 the following holds:
If a1 , a2 , . . . , an Z such that each ai 1 (mod 4), then a1 a2 . . . an (mod 4).
=
(2) Generalization of the innitude of primes.
(a) Prove tha