Welcome to Math 240! (additions to this web site no longer count towards good deed points)

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Week of...

Notes and Links

1

Sep 8

About This Class, What is this class about? (PDF, HTML), Monday, Wednesday

2

Sep 15

HW1, Monday, Wednesday, TheComplexField.pdf,HW1_solutions.pdf

3

Sep 22

HW2, Class Photo, Monday, Wednesday, HW2_solutions.pdf

4

Sep 29

HW3, Wednesday, Tutorial, HW3_solutions.pdf

5

Oct 6

HW4, Monday, Wednesday, Tutorial, HW4_solutions.pdf

6

Oct 13

No Monday class (Thanksgiving), Wednesday, Tutorial

7

Oct 20

HW5, Term Test at tutorials on Tuesday, Wednesday

8

Oct 27

HW6, Monday, Why LinAlg?, Wednesday, Tutorial

9

Nov 3

Monday is the last day to drop this class, HW7, Monday, Wednesday, Tutorial

10

Nov 10

HW8, Monday, Tutorial

11

Nov 17

MondayTuesday is UofT November break

12

Nov 24

HW9

13

Dec 1

Wednesday is a "makeup Monday"! EndofCourse Schedule, Tutorial

F

Dec 8

The Final Exam

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Boris
Problem
Find a set $S$ of two elements that satisfies the following:
(1) $S$ satisfies all the properties of the field except distributivity.
(2) $\exists x\in S,0x\neq 0$.
Solution:
Let $S=\{a,b\}$ where $a$ is the additive identity and $b$ is the multiplicative identity and $a\neq b$. After trial and error, we have the following addition and multiplication tables:
$+$

$a$

$b$

$a$

$a$

$b$

$b$

$b$

$a$

$\times$

$b$

$a$

$b$

$b$

$a$

$a$

$a$

$b$

We verify that $S$ satisfies (1). By the addition and multiplication tables, then $S$ satisfies closure, commutativity, associativity and existence of identities and inverses. Since $a(b+b)=a(a)=b\neq a=a+a=ab+ab$, then $S$ does not satisfy distributivity. Then $S$ satisfies (1).
We verify that $S$ satisfies (2). Since $aa=b\neq a$, then $S$ satisfies (2).
Elementary Errors in Homework
(1) Prove $A\implies B$. Assume $A$ and derive $B$. It is not the other way around.
(2) Prove $A\iff B$. Show that $A\implies B$ and $B\implies A$.
(3) This is for Boris's section only. When a proof requires a previous result, there are two possibilities:
 (a) The result is already proved in class or in a previous homework. Then state the result and use it without proof.
 (b) The result is neither proved in class nor in a previous homework. Then reference it in the textbook or prove it yourself.
Nikita