Math 311W Problem Set 2
Due Friday, September 9, 2016
1. Define f : R R by f (x) = 2x + 1. Is f one-to-one (injective) or onto (surjective)?
(Be sure to prove your answers.)
Solution. 1) one-to-one: take x1 and x2 , and assume f (x1 ) = f (x2 ). Then
2x1
Part 1: Summaries of Special Aspects of Writing Mathematics Papers
Categories of Mathematics Paper
In the section 2 What Kind of Mathematics Paper, the author states two different kind of main
mathematics paper: research mathematics paper and expository m
Part 1: Summaries of Special Aspects of Writing Mathematics Papers
Categories of Mathematics Paper
In Section 2, What Kind of Mathematics Paper, the author states that there are two main
different kinds of mathematical paper: research mathematics paper an
A
T
T
T
T
F
F
F
F
B
T
T
F
F
T
T
F
F
C
T
F
T
F
T
F
T
F
not A
F
F
F
F
T
T
T
T
not B
F
F
T
T
F
F
T
T
not C
F
T
F
T
F
T
F
T
BC
T
F
F
F
T
F
F
F
(notB)v(notC)
F
T
T
T
F
T
T
T
A->(B C)
T
F
F
F
T
T
T
T
(notB)v(notC)->(notA)
T
F
F
F
T
T
T
T
AvB
T
T
T
T
T
T
F
F
A->
Bonus:
Theorem: The second principle of mathematical induction is logically equivalent to the wellordering principle for the set N of the natural numbers.
Proof:
A. If the Second Principle of Mathematical induction is true, then the well-ordering
principl
Math 311W Problem Set 4
Due Friday, September 23, 2016
1. Find gcd(a, b) and express the gcd as a linear combination of a and b.
(a) a = 361, b = 1178.
(b) a = 525, b = 231.
Solution. (a) Applying the Euclidean algorithm, we obtain
1178 = 361 3 + 95
361 =
Math 311W Problem Set 1
Due Friday, September 2, 2016
Please write up your solutions in complete sentences, and include enough detail so that a
fellow student can follow your arguments.
1. Prove that the sum of two odd integers is even.
Proof. Let a and b
Math 311W Problem Set 3
Due Friday, September 18, 2015
1. Let f : R R be a function defined by f (x) = 2x + 1. Find the inverse of f . (Be sure
to give an explicit formula.)
Solution. Let y = 2x + 1. To find f 1 , we solve y = 2x + 1 for x. Then
x=
So f 1
Math 311W Problem Set 2
Due Friday, September 9, 2016
1. Define f : R R by f (x) = 2x + 1. Is f one-to-one (injective) or onto (surjective)?
(Be sure to prove your answers.)
2. Let f : Z Z be a function defined by f (x) = x2 . Is f one-to-one (injective
Math 311W Problem Set 1
Due Friday, September 2, 2016
Please write up your solutions in complete sentences, and include enough detail so that a
fellow student can follow your arguments.
1. Prove that the sum of two odd integers is even.
2. Prove that if t
Math 311W Problem Set 3
Due Friday, September 16, 2016
1. Let f : R R be a function defined by f (x) = 2x + 1. Find the inverse of f . (Be sure
to give an explicit formula.)
2. Let f : R2 R2 be a function defined by f (x, y) = (x + y, x y). Find the inver
1.
1. If A, then B
1. If a function is continuous, then that function can be written as a sum of integer
powers.
2. Let n be a counting number. If an integer is in form of n2-1, then that integer is
not prime.
3. If n is composite, then the number 2n-1 is