Math 311W Problem Set 7
Due Friday, October 30, 2015
1. Solve the following simultaneous congruences.
(a)
(
3x 1 mod 5
2x 4 mod 11
Solution. Since 3 2 1 mod 5, 3x 1 mod 5 becomes
x2
mod 5.
x2
mod 11.
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 Mathemat
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)
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 kin
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
mathema
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
n
Math 311W: Discrete Math
Handout, Friday, April 7
5.1 Preliminaries of group theory
1. Cancellation and its consequences.
Theorem 5.1.1. Let G be a group and let a and b be elements of G. Then there a
Math 311W: Discrete Math
Handout, Wednesday, April 5
4.2 The order and sign of a permutation.
1. The sign of a permutation: a geometric interpretation. To motivate the concept
of the sign of a permuta
Math 311W: Discrete Math
Handout, Monday, April 3
4.2 (The order and sign of a permutation), with part of 5.1.
1. The order of a group element.
Notation: Powers of an element in the group. Let g be a
Math 311W: Discrete Math
Handout, Friday, March 3, & Monday, March 13
2.3 Relations. Relations can be viewed as a generalization of functions, where
we now allow one input to correspond to multiple ou
Math 311W: Discrete Math
Handout, Wednesday, March 15
3.1 Propositional Logic. Logic lays the foundation for modern math. It is what
makes mathematical statements precise and their proofs rigorous, wh
Math 311W: Discrete Math
Handout, Monday, Feb. 20
2.1 Elementary Set Theory. The concept of sets plays a fundamental role in
the language of mathematics.
1. A set is a collection of objects, known as
Math 311W: Discrete Math
Handout, Wednesday, March 15
3.1 Propositional Logic. Logic lays the foundation for modern math. It is what
makes mathematical statements precise and their proofs rigorous, wh
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
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
Math 311W Problem Set 6
Due Friday, October 23, 2015
1. Let a, b Z such that a b mod n. If m is a positive integer such that m | n, then
prove that
a b mod m.
Solution. Let n = md for some d.
Let a =
Math 311W Problem Set 3
Due Friday, September 18, 2015
1. Let f : R R be a function defined by f (x) = x5 + 1. Find the inverse of f . (Be sure
to give an explicit formula.)
Solution. Let y = x5 + 1.
Math 311W Problem Set 1
Due Friday, September 4, 2015
Please write up your solutions in complete sentences, and include enough detail so that a
fellow student can follow your arguments.
1. Prove that
Math 311W Problem Set 2
Due Friday, September 11, 2015
1. Let f : Z Z be a function defined by f (x) =
(surjective)? (Be sure to prove your answers.)
x
2 . Is f one-to-one (injective) or onto
Solution
Math 311W Problem Set 4
Due Friday, September 25, 2015
1. Let a, b, c Z with a 6= 0. Prove that if a | b and a | c, then a | (bm + cn) for any
n, m Z.
Solution. Since a | b and a | c, we know b = ak1
Math 311W Problem Set 5
Due Friday, October 16, 2015
1. Find the prime factorizations for 51 and 198, and use your answers to find the gcd of
51 and 198. (No proof is required here, just write down th
Math 311W Problem Set 8
Due Friday, November 20, 2015
You may use calculator for this homework.
1. You and your friend Kay wish to agree on a secret key using the Diffie-Hellman key
exchange. Kay anno
Math 311W Problem Set 9
Due Friday, December 4, 2015
1. (9 pts) Determine which of the following (G, ) are groups:
(a) G = cfw_a + b 5 | a, b Z and the operation is addition;
(b)
G=
1 a
a 1
| aR
and t
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: 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 func
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
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 func
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.
ix; X: Y: cfw_of cfw_j .2 73: gtswjfgwgljf
at in) I.
2 er, .va /. M a
the] o 1
Math 311W: Discrete Math
Handout, Friday, March 3, 85 Monday, March 13
2.3 Relations. Relations can be viewed as a gene