14-240/Tutorial-Sep30

DBN: Classes: 2014-15: 14-240 / Navigation Welcome to Math 240! (additions to this web site no longer count towards good deed points) # 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 Monday-Tuesday is UofT November break 12 Nov 24 HW9 13 Dec 1 Wednesday is a "makeup Monday"! End-of-Course Schedule, Tutorial F Dec 8 The Final Exam Register of Good Deeds Add your name / see who's in!

Boris

Problem

Find a set [math]\displaystyle{ S }[/math] of two elements that satisfies the following:

(1) [math]\displaystyle{ S }[/math] satisfies all the properties of the field except distributivity.

(2) [math]\displaystyle{ \exists x \in S, 0x \neq 0 }[/math].

Solution:

Let [math]\displaystyle{ S = \{ a, b \} }[/math] where [math]\displaystyle{ a }[/math] is the additive identity and [math]\displaystyle{ b }[/math] is the multiplicative identity and [math]\displaystyle{ a \neq b }[/math]. After trial and error, we have the following addition and multiplication tables:

[math]\displaystyle{ + }[/math]	[math]\displaystyle{ a }[/math]	[math]\displaystyle{ b }[/math]
[math]\displaystyle{ a }[/math]	[math]\displaystyle{ a }[/math]	[math]\displaystyle{ b }[/math]
[math]\displaystyle{ b }[/math]	[math]\displaystyle{ b }[/math]	[math]\displaystyle{ a }[/math]

[math]\displaystyle{ \times }[/math]	[math]\displaystyle{ b }[/math]	[math]\displaystyle{ a }[/math]
[math]\displaystyle{ b }[/math]	[math]\displaystyle{ b }[/math]	[math]\displaystyle{ a }[/math]
[math]\displaystyle{ a }[/math]	[math]\displaystyle{ a }[/math]	[math]\displaystyle{ b }[/math]

We verify that [math]\displaystyle{ S }[/math] satisfies (1). By the addition and multiplication tables, then [math]\displaystyle{ S }[/math] satisfies closure, commutativity, associativity and existence of identities and inverses. Since [math]\displaystyle{ a(b + b) = a(a) = b \neq a = a + a = ab + ab }[/math], then [math]\displaystyle{ S }[/math] does not satisfy distributivity. Then [math]\displaystyle{ S }[/math] satisfies (1).

We verify that [math]\displaystyle{ S }[/math] satisfies (2). Since [math]\displaystyle{ aa = b \neq a }[/math], then [math]\displaystyle{ S }[/math] satisfies (2).

Elementary Errors in Homework

(1) Prove [math]\displaystyle{ A \implies B }[/math]. Assume [math]\displaystyle{ A }[/math] and derive [math]\displaystyle{ B }[/math]. It is not the other way around.

(2) Prove [math]\displaystyle{ A \iff B }[/math]. Show that [math]\displaystyle{ A \implies B }[/math] and [math]\displaystyle{ B \implies A }[/math].

(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 the textbook if the proof is in the textbook or prove it yourself.