Preparing for proof-based mathematics at university Watch

Lockie123
Badges: 0
Rep:
?
#1
Report Thread starter 4 years ago
#1
Once I've finished AS/A-level mathematics and further mathematics, how can I prepare for proof-based mathematics at university? I will be doing calculus and linear algebra in first year.
0
reply
0x2a
Badges: 2
Rep:
?
#2
Report 4 years ago
#2
For an introduction to proofs you can look at any introductory textbook on (Naive) Set Theory or Elementary Logic. They'll teach you operations defined on sets, the various problems associated with naive set theory, a look at logical quantifiers and negation and methods of proof like contradiction, contraposition and induction. Books like "How to Prove it" by Velleman and "Book of Proof" by Hammack seem to be commonly recommended texts as well.

If you want to get a look at proof-based calculus or linear algebra then "Calculus" by Spivak will be more than enough for the former, and something like "Finite Dimensional Vector Spaces" by Halmos or "Linear Algebra Done Right" by Axler will be great for the latter.

If you're taking analysis in the first year then I'd recommend reading "Understanding Analysis" by Abbott. It's an excellent textbook although I'm using it just as a supplement at the moment.
0
reply
Lockie123
Badges: 0
Rep:
?
#3
Report Thread starter 4 years ago
#3
(Original post by 0x2a)
For an introduction to proofs you can look at any introductory textbook on (Naive) Set Theory or Elementary Logic. They'll teach you operations defined on sets, the various problems associated with naive set theory, a look at logical quantifiers and negation and methods of proof like contradiction, contraposition and induction. Books like "How to Prove it" by Velleman and "Book of Proof" by Hammack seem to be commonly recommended texts as well.

If you want to get a look at proof-based calculus or linear algebra then "Calculus" by Spivak will be more than enough for the former, and something like "Finite Dimensional Vector Spaces" by Halmos or "Linear Algebra Done Right" by Axler will be great for the latter.

If you're taking analysis in the first year then I'd recommend reading "Understanding Analysis" by Abbott. It's an excellent textbook although I'm using it just as a supplement at the moment.
What AS/A-Level maths and further maths topics do I need to know well before moving on to books such as "How to Prove It" and "Understanding Analysis"?

What order should I use all of those 4 books in? Should I read "How to Prove It" and "Understanding Analysis" before moving on to "Calculus" by Spivak and "Linear Algebra Done Right'?
0
reply
0x2a
Badges: 2
Rep:
?
#4
Report 4 years ago
#4
(Original post by Lockie123)
What AS/A-Level maths and further maths topics do I need to know well before moving on to books such as "How to Prove It" and "Understanding Analysis"?

What order should I use all of those 4 books in? Should I read "How to Prove It" and "Understanding Analysis" before moving on to "Calculus" by Spivak and "Linear Algebra Done Right'?
You'll definitely need to know all of C1-C4, and I'd say most if not all of the Further Pure stuff simply because you need the mathematical maturity to read these books. You really aren't going to require the knowledge of knowing how to solve differential equations, but you need to be able to think mathematically. If you've done any BMO or STEP papers, those are good too.

"How to Prove it" is probably the most basic book in that list, and it'll teach you about the logic and set theory I was talking about. Between "Understanding Analysis" and "Calculus", I'd say sometimes "Calculus" is the more difficult book (in terms of the problems) but "Understanding Analysis" deals with a more rigorous treatment of single variable calculus (real analysis), so it really depends upon what kind of course you'll be taking.
And you can read through "Linear Algebra Done Right" whenever, although the abstract nature of the text (it does not deal with applications, nor does it even deal with determinants) might throw you off a bit.
0
reply
tombayes
  • Study Helper
Badges: 12
Rep:
?
#5
Report 4 years ago
#5
(Original post by Lockie123)
Once I've finished AS/A-level mathematics and further mathematics, how can I prepare for proof-based mathematics at university? I will be doing calculus and linear algebra in first year.
Martin Liebeck: A concise introduction to pure mathematics

What other maths courses are you doing because analysis is the real proof based one, calculus sounds applied and linear algebra is in the middle.

Also, you would probably do an introduction to proofs course like I think mine was called 'numbers and sets' if i remember correctly.
0
reply
Lockie123
Badges: 0
Rep:
?
#6
Report Thread starter 4 years ago
#6
(Original post by 0x2a)
You'll definitely need to know all of C1-C4, and I'd say most if not all of the Further Pure stuff simply because you need the mathematical maturity to read these books. You really aren't going to require the knowledge of knowing how to solve differential equations, but you need to be able to think mathematically. If you've done any BMO or STEP papers, those are good too.

"How to Prove it" is probably the most basic book in that list, and it'll teach you about the logic and set theory I was talking about. Between "Understanding Analysis" and "Calculus", I'd say sometimes "Calculus" is the more difficult book (in terms of the problems) but "Understanding Analysis" deals with a more rigorous treatment of single variable calculus (real analysis), so it really depends upon what kind of course you'll be taking.
And you can read through "Linear Algebra Done Right" whenever, although the abstract nature of the text (it does not deal with applications, nor does it even deal with determinants) might throw you off a bit.
I'll definitely start with 'How to Prove it'. Then I'll move on to 'Understanding Analysis', 'Calculus' and 'Linear Algebra Done Right'.

Thank you!
0
reply
Lockie123
Badges: 0
Rep:
?
#7
Report Thread starter 4 years ago
#7
(Original post by tombayes)
Martin Liebeck: A concise introduction to pure mathematics

What other maths courses are you doing because analysis is the real proof based one, calculus sounds applied and linear algebra is in the middle.

Also, you would probably do an introduction to proofs course like I think mine was called 'numbers and sets' if i remember correctly.
Could you tell me what you mean by this?
0
reply
tombayes
  • Study Helper
Badges: 12
Rep:
?
#8
Report 4 years ago
#8
(Original post by Lockie123)
Could you tell me what you mean by this?
are you planning to study mathematics in its own right or another subject like engineering or physics?
0
reply
Lockie123
Badges: 0
Rep:
?
#9
Report Thread starter 4 years ago
#9
(Original post by tombayes)
are you planning to study mathematics in its own right or another subject like engineering or physics?
Engineering. Is it better for me to use proof-based textbooks or skill-based textbooks (which focus more on applying the knowledge)?
0
reply
tombayes
  • Study Helper
Badges: 12
Rep:
?
#10
Report 4 years ago
#10
(Original post by Lockie123)
Engineering. Is it better for me to use proof-based textbooks or skill-based textbooks (which focus more on applying the knowledge)?
Engineering will not have any (serious) proof based maths. Just make sure you are comfortable with A level+FM material
0
reply
Lockie123
Badges: 0
Rep:
?
#11
Report Thread starter 4 years ago
#11
(Original post by tombayes)
Engineering will not have any (serious) proof based maths. Just make sure you are comfortable with A level+FM material
Is there any benefit for an engineering student to go through proof-based mathematics?

Also, would you say that GCSE mathematics, AS/A-Level mathematics and further mathematics are more proof-based or skill-based?
0
reply
tombayes
  • Study Helper
Badges: 12
Rep:
?
#12
Report 4 years ago
#12
(Original post by Lockie123)
Is there any benefit for an engineering student to go through proof-based mathematics?
well there is a benefit in anyone learning about proof and logic. But for engineering you certainly will not need to know/use proof. In fact I am certain you do not cover any serious proofs at all.
0
reply
Lockie123
Badges: 0
Rep:
?
#13
Report Thread starter 4 years ago
#13
(Original post by tombayes)
well there is a benefit in anyone learning about proof and logic. But for engineering you certainly will not need to know/use proof. In fact I am certain you do not cover any serious proofs at all.
I am the type of person who really needs to understand things in order to remember anything which is why I thought focusing on proofs would be a better idea than skill-based mathematics. What do you think?

Also, would you say that GCSE mathematics, AS/A-Level mathematics and further mathematics are more proof-based or skill-based?
0
reply
tombayes
  • Study Helper
Badges: 12
Rep:
?
#14
Report 4 years ago
#14
(Original post by Lockie123)
I am the type of person who really needs to understand things in order to remember anything which is why I thought focusing on proofs would be a better idea than skill-based mathematics. What do you think?

Also, would you say that GCSE mathematics, AS/A-Level mathematics and further mathematics are more proof-based or skill-based?
entirely skill-based the only proof technique you learn is induction (further maths) and its a trivial technique. You do not need proofs, trust me. For example, something 'easy' like \int_0^1 x^3 dx takes about a year of material (for those just studying maths a uni) to be able to prove rigorously (look up Riemann integration if your interested). Clearly not helpful at A-Level, Further maths and before.
0
reply
Smack
  • TSR Support Team
Badges: 20
Rep:
?
#15
Report 4 years ago
#15
(Original post by Lockie123)
Is there any benefit for an engineering student to go through proof-based mathematics?
Not really, no, I think there are far more useful things to spend your time doing to compliment your studies or help with your employability, but as an endeavour it's more productive than lots of other things you could be doing, e.g. wasting time on the internet.
0
reply
Lockie123
Badges: 0
Rep:
?
#16
Report Thread starter 4 years ago
#16
(Original post by Smack)
Not really, no, I think there are far more useful things to spend your time doing to compliment your studies or help with your employability, but as an endeavour it's more productive than lots of other things you could be doing, e.g. wasting time on the internet.
Thank you!
0
reply
X

Quick Reply

Attached files
Write a reply...
Reply
new posts
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise

University open days

  • Cardiff University
    Undergraduate Open Day Undergraduate
    Wed, 27 Mar '19
  • University of Portsmouth
    Postgraduate and Part-Time Open Evenings Postgraduate
    Wed, 27 Mar '19
  • Middlesex University London
    Postgraduate Open Evening Postgraduate
    Wed, 27 Mar '19

How old were you when you first saw porn?

I've never seen it (144)
20.57%
Before I was 12 (238)
34%
13 (110)
15.71%
14 (84)
12%
15 (44)
6.29%
16 (22)
3.14%
17 (9)
1.29%
18 (6)
0.86%
Between the ages of 19 - 24 (11)
1.57%
Over 25 (2)
0.29%
12 (30)
4.29%

Watched Threads

View All