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MAT246: CONCEPTS IN ABSTRACT MATHEMATICS

创建日期：2024-05-20 16:52:49

所属分类：数学mathematics

标签： MAT246

CONCEPTS IN ABSTRACT MATHEMATICS

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MAT246: CONCEPTS IN ABSTRACT

MATHEMATICS
week 5
TERM TEST 1 THIS FRIDAY:
Test 1 Friday Feb. 10, 5-7 pm. various locations; information posted
Early seating: 3:30-5:00, BA 1210
Details will be communicated to those who apply (later this evening, Wed.)
Help hours: myself Tues and Thurs 3:30-4:30,
Oliver: Wed. 1:30-2:30 zoom
Aidan: Friday (see announcement) zoom
Esmeralda: Friday PG 2:30-3:30
WHAT IS A ROW OF THE MULTIPLICATION TABLE?
Which is the row in front 2 ?
Which is the row corresponding to 3?
The nature of these rows are discussed in the
tutorial document.
And will also appear in the test.
FEEDBACKS ARE OUT
General comment:
“Approaches are too mechanical” (not enough soul, not enough intuitive understanding) , and
“often people have memorized a cliché argument, from beginning to end” (lack of analytical thinking)
Lecture quiz 2: for a given > 1, assume all natural numbers 1 < < have a prime divisor, prove must have a prime divisor as well.
Marker’s comments:
“Probably at least half went through an entire induction proof, and proved the statement for n+1
rather than n. Some were trying to use the statement for n to prove it for n+1, which no one did
successfully. That's why they need complete induction... ”
PMI VS PCMI
PS1 Q7 was specifically asking to use PCMI, but majority did make the assumption of
PCMI, but didn’t use it; instead, they presented a proof by PMI.
The proof of PMI was harder and longer, less intuitive, and common in the literature,
so many students’ proofs were similar !
The power of PCMI was tested in Q7: a stronger principle can make the argument a
bit shorter, more intuitive and more efficient.
PS1 QUESTIONS 3C AND 4B
Q 3c:
“Students needed to recognize that one cannot use properties of normal arithmetic on
fractions because those properties rely in this very proof, as well as the next few parts.
So students who “divided by m on both sides” or attempted a similar operation without
reference to a cancellation law lost marks. The most common error, however, was failing
to cite the UNIQUENESS of the solution n/m to the equation mx=n. Without this
uniqueness you cannot conclude that two solutions to the equation are equal.”
Q 4b: “most people eliminated 0,2,4,6,8,5. Many tried to eliminate 1 and 9 separately,
I think it is hard to do that clearly, more or less they lost a portion of marks for logical
errors.”
COMMENTS ON Q2 AND Q8 ARE YET TO ARRIVE
CONCLUSION OF FEEDBACK COMMENTS
The plan is to keep on proving very easy facts as part of a `theory toolbox’ in order to:
1. prove ideas, and to add them as theorems to enrich the theory
2. apply theory to simplify computation (be aware of smallest pieces of `theory’; they
form a logical structure for making decision at the time of computation.)
3. gradually connecting our arguments to our intuitive knowledge (the soul of our
knowledge)
Please carefully read the questions, understand the logical structure, and answer as
simple and as focused as possible by using your own intuitive knowledge, and
Avoid the desire to make things complicated: memorizing big structures and mindlessly
copying them down as answers, thinking that you have written something formal. The
proofs must be yours!
TYPE OF UNIVERSITY EDUCATION
Longer lasting Skills vs memorizing well prepared packages of information
Reflective and analytical skills
Theory building, and using of theory for practical problem solving
Gradually connecting the intuitive knowledge to formal presentation
This means establishing a healthy channel between
‘the thinking you’ and ‘your audience’
TERM TEST 1: EXPECTATIONS
Type of questions:
short arguments, very close to the definitions, statements of the theorems, and basic
properties; make sure to have a practical understanding of definitions and basic
facts about various concepts.
Steps of a proof are asked in an earlier question, so be aware of the whole page,
not a single question.
Arguments cut from a larger proof; a larger argument is made of smaller pieces;
learn them that way.
use of theory to do computations instead of brute force.
Theory is being developed in the same paper; use it.
THEME OF THIS WEEK’S TUTORIALS: SOLVING EQUATIONS
Solving ≡ ( ):
Infancy: (chapter 3) use multiplication table! This way of solving equation is by consulting the
multiplication tables.
High school: using basic Algebra:
Cancellation Law: (chapter 4, for mod prime) solve the equation by repeated cancellation, and
repeatedly changing numbers in mod so that the cancellation becomes apparent. Eventually,
and in stages, cancel . Eg: solve 5 ≡ 17 ( 19)
University math: Field operations and properties:
Using multiplicative inverse: (chapters 4, 5, and still in mod prime) this could also include
applications of Fermat’s theorem in chapter 5.
More Advance: Reducing the strong field assumptions
Relatively prime: (Chapter 7, mod non-prime) Euler’s ideas and using Diophantine equations.

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