MATH 145 Algebra, Lecture Notes
by Stephen New
Chapter 0: Logic and Proof
1. Introduction
1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs
contained paradoxes, and t
Chapter 2. Rings, Fields, Orders and Induction
o see complete
otation on Ordered
d Axioms
e by first order
age on Onenote
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2.1 Definition: A ring (with identity) is a set R with elements
A VERY BRIEF INTRODUCTION TO CRYPTOGRAPHY
1. Primality Test
Given any positive integer n, is there a way to tell if n is a prime or not? Is
there an
efficient way to tell if n is a prime or not? Of c
MATH 145 Algebra, Solutions to Assignment 4
1: (a) Find the inverse of 178 in Z365 .
Solution: We find s and t so that 178s + 365t = 1, and then 1781 = s. The Euclidean Algorithm gives
365 = 2 178 + 9
a cfw_1, 2.,n->A
functioncfw_1, 2.,n->A
functionakka
akA
Chapter 2. Recursion and Induction
2.1 Definition: Let n be a positive integer and let A be a set. An ordered n-tuple with
entries in A is a f
Chapter 5. Complex Numbers
5.1 Definition: A complex number is a vector in R2 . The complex plane, denoted
by C, is the set of complex numbers:
x
2
C=R =
x R, y R .
y
0
1
0
x
0
In C we u
MATH 145 Algebra, Solutions to Assignment 2
1: Let (an )n0 be the Fibonacci sequence, so a0 = 0, a1 = 1 and an = an1 + an2 for n 2.
(a) Show that an1 an+1 = an 2 + (1)n for all n 1.
Solution: When n =
MATH 145 Algebra, Solutions to Assignment 5
1: Solve each of the following equations for z C. Express your answers in cartesian form.
(a) i z 2 + (2 + i) z + (7 + i) = 0
Solution: By the Quadratic For
MATH 145 Proof Rules
Basic Truth-Equivalences
For any formulas F , G and H, any terms s and t, and any variable symbols x and y, we have the following basic
truth-equivalences.
(Identity)
(Double Nega
MATH 145 Algebra, Assignment 3
Due: Wed Nov 16
1: Let a = (25)! and b = (5500)3 (1001)2 .
(a) Find the prime factorization of a and of b.
(b) Find the prime factorization of gcd(a, b) and of lcm(a, b)
MATH 145 Algebra, Ordered Field Axioms
Does not have to be
and multiplying, ca
binary function we
Definition: A ring (with identity) is a set R with two distinct elements 0, 1 R, called
the zero and i
MATH 145 Algebra, Solutions to Assignment 8
1: (a) Find 17458 mod 13.
Solution: Since 17 = 4 mod 13 we have 17458 = 4458 mod 13. By Fermats Little Theorem, the list of powers
of 4 modulo 13 repeats ev
of on hand
H 145
016-10-21
K7
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MATH 145
6-10-25
Chapter 3. Factorization of Integers
2.1 Definition: For a, b Z we say that
a divides b (or that a is a factor of b, or that
b is a multiple of a), an
MATH 145 Algebra, Solutions to the Midterm Test, Fall 2016
[10]
1: (a) Let F = (P R) (Q R) and G = (Q R) P ) (P R) .
Determine whether F G.
Solution: We make a truth-table for
P Q R P R
1 1 1
1
1 1 0
MATH 145 Algebra, Assignment 6
Due Fri Nov 13
1: For each of the following pairs (a, b), use the Euclidean Algorithm with Back-Substitution
to nd d = gcd(a, b) and to nd integers s and t such that as
MATH 145 Algebra, Assignment 5
Due Fri Oct 30
1: The axioms for a ring and for an ordered eld can be found on the following page.
(a) Let F be an ordered eld. Using only the 14 axioms which dene an or
MATH 145 Algebra, Assignment 4
Not to hand in
In this assignment, a is a constant symbol; x, y and z are variable symbols; f , g and h are
function symbols with f unary and g and h binary; and r is a
MATH 145 Algebra, Assignment 3
Due: Fri Oct 9
1: Make a derivation for each of the following equivalences. Provide justication at each step.
(a) (P Q) (P R) P R
(b) (P Q) R R P P
(c) (P (Q R) (P Q) (P
MATH 145 Algebra, Assignment 2
Due: Fri Sept 25
1: Let S = cfw_1, 2, 3, 4, 5, 6, 7, 8, 9, 10. List all of the elements in each of the following sets.
(a) A = x S x is even or x is a multiple of 3
(b)
MATH 145 Algebra, Assignment 1
Due: Fri Sept 18
x
1
.
=
1+x
3 x
with 0 < x 1 for x in terms of y.
1: (a) Find all real numbers x such that
(b) Solve y = x +
1
x
2: (a) Find all ordered pairs of integ
MATH 145 Algebra, The Euclidean Algorithm with Back Substitution
Theorem: (The Euclidean Algorithm With Back-Substitution) Let a and b be integers
and let d = gcd(a, b). Then there exist integers s an
MATH 145 Algebra (Advanced Level), Course Outline, Fall 2015
Lectures: Mon, Wed, Fri 12:30-1:20 in MC 2038, and Thurs 12:30-1:20 in MC 4059.
Instructor: Stephen New, tel: 35554, oce: MC 5419, hours: M
MATH 145 Algebra (Advanced Level), Course Outline, Fall 2017
Lectures: Section 1: M, W, F 10:30-11:20 in STC 50 and Tu 4:30-5:20 in MC 4059.
Section 2: M, W, F from 8:30-9:20 in in STC 20 and Tu 8:30-
MATH 145 Algebra, Assignment 1
Due: Wed Oct 5
1: Solve each of the following problems by making a truth-table.
(a) Determine whether |= (P (Q R) (Q R) (P Q) .
(b) Determine whether (P Q) R) Q (Q R) (P
MATH 145 Algebra, Assignment 4
Due: Wed Nov 30
1: (a) Find the inverse of 178 in Z365 .
(b) Solve the linear congruence 356 x = 28 mod 730.
654
(c) Find 523470
mod 37.
(d) Solve the pair of congruence
MATH 145 Algebra, Assignment 1
Due: Fri Oct 6
1: Solve each of the following problems by making a truth-table.
(a) Determine whether |= (P Q) R) (R (P Q) .
(b) Determine whether (P R) (Q R) (P Q) (Q R