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A Summer of Maths (ASoM) 2016

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Btw, when you want to show that the set

\mathbb{Z}

is a group with respect to addition, do you have to show that the set satisfies all the four properties of a group (with respect to a binary operation) or can we only prove property (3)

You need to prove all four, but for something as trivial as integers under addition, the other three are so trivial that he didn't bother proving it. Every element a has an inverse (-a), closed because sum of two integers is an integer, obviously associative. There, three properties done in one line.

Reply 221

Insight314

Original post by Zacken

1) His statement about having to prove property (3) being enough to categorise a group has nothing to do with proving that set

Z

with respect to operation addition is a group. My question is about Exercise 1.2 question 1, and his statement was two pages before when he was discussing the Definition of a group.

2)I know it is trivial and can be done in a few lines (non-rigorously in a few lines) but my question is whether proving property 3 automatically proves that the set is a Group. I would be surprised if it was, but I am confused by his statement before. In context, he is generalising G to be a group and saying that it is enough to prove the existence of an identity element.

Reply 222

username1162972

Original post by Insight314

k m8, I am here having fun doing Groups with Beardon and chilling with the fact I do Groups and V&M at the same time.

Btw, when you want to show that the set

\mathbb{Z}

is a group with respect to addition, do you have to show that the set satisfies all the four properties of a group (with respect to a binary operation) or can we only prove property (3) i.e "there is a unique

e

G

such that for all

g

G

g*e = e = e*g

" ? I am currently doing one of the exercises in the textbook and I have gone so much in detail on proving all the four properties, whereas Beardon states that "It follows that when we need to prove that, say,

G

is a group we need only prove" property (3).

Yeh basically what zain said.
Try proving Z is a group under subtraction operation, see what goes wrong.

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Reply 223

username1162972

Original post by Insight314

1) His statement about having to prove property (3) being enough to categorise a group has nothing to do with proving that set

Z

To your 2nd question na. Always prove all 4 for proper groups he just missed it here due to what zain said, cos its easy.

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Reply 224

Zacken

Original post by Insight314

1) His statement about having to prove property (3) being enough to categorise a group has nothing to do with proving that set

Z

Rigorously in a few lines as well, unless you write really big. And no, it's fairly obvious that what he's saying by that quote is that you need not prove the uniqueness of

e

every time you need to prove an object is a group, rather you need to prove the existence of

e

and uniqueness always follows immediately. That is, he is not saying that proving

e

exists proves the object is a group. Rather, you need to prove the three axioms, then when proving that

e

exists and is unique, it suffices to prove that

e

exists only, you need not prove uniqueness. The uniqueness of

e

follows immediately from the existence as he's proven in that paragraph.

(edited 7 years ago)

Reply 225

Insight314

Original post by physicsmaths

Yeh basically what zain said.
Try proving Z is a group under subtraction operation, see what goes wrong.

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He didn't answer my question though?

Property (4) is not satisfied under subtraction operation because then f and g must be equal in order for

g*h = e = h*g

. This is in contrary to the addition operation, where you have

\mathbb{Z^{-}}

and

\mathbb{Z^{+}}

being inverse sets of each other.

Reply 226

Insight314

Original post by Zacken

Rigorously in a few lines as well, unless you write really big. And no, it's fairly obvious that what he's saying by that quote is that you need not prove the uniqueness of

e

every time you need to prove an object is a group, rather you need to prove the existence of

e

and uniqueness always follows immediately. That is, he is not saying that proving

e

exists proves the object is a group. Rather, you need to prove the three axioms, then when proving that

e

exists and is unique, it suffices to prove that

e

exists only, you need not prove uniqueness. The uniqueness of

e

follows immediately from the existence as he's proven in that paragraph.

I see now, I thought he was referring to proving that a set is a group and not of the uniqueness of

e

. I did realise that, but thought that he was also saying that proving property (3) immediately proves the other ones which makes no sense whatsoever.

I still think you can't prove it 'rigorously' in one line like you said before, or at least you need some mathematical writing; and not just state blatantly which property is satisfied, but also show how it is satisfied, although like you said it is very trivial.

Reply 227

sweeneyrod

Anyone doing N&S? I think I'll do a bit of that and a bit of Groups. Also, a useful resource -- Gowers' blog (select category, Cambridge teaching).

Reply 228

ln(sec(x))

Sorry if this question has already been answered (I had a quick scan through and couldn't find anything), but does anyone have any recommendations for where to start in all this? Is it a case of dotting between resources as I feel like it, or is there any suggested order to learning some of Numbers and Sets, Groups, and Vectors and Matrices? :smile:

Reply 229

Zacken

Original post by ln(sec(x))

I'd advise with starting with Numbers and Sets, it's the course that gets you used to thinking about uni maths properly (in terms of proofs, a proper definition of functions and how they work, what sets are...),with not much new content. Win-win.

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Reply 230

Zacken

Original post by Insight314

I still think you can't prove it 'rigorously' in one line like you said before, or at least you need some mathematical writing; and not just state blatantly which property is satisfied, but also show how it is satisfied, although like you said it is very trivial.

This works:

a, b \in \mathbb{Z} \Rightarrow (a+b) \in \mathbb{Z}

so closure, also

(a+b) + c = a + b + c = a + (b+c)

so associativity. Note

a + 0 = 0 + a = a

so identity and

a + (-a) = (-a) + a = 0

so inverse.

Reply 231

EricPiphany

Tentatively proposed errata for the zeroth and first chapter of the notes here: https://dec41.user.srcf.net/notes/IA_M/groups.pdf

Page 4, Now we are studying...

rs^2

should be

r^2s

?

Page 7, Suppose e and... treating e as an inverse should be treating e as an identity? same for e'
Page 7, Proof: Given...

ax^{-1}

should be

a^{-1}x

Page 9, Proof that (i)... duplicate from beginning of proof.
Page 9, Otherwise, suppose... a does not divide n should read n does not divide a.

Page 13, Definition (order... smallest integer, shouldn't that be smallest positive integer? same on last line.

Page 14, Definition (direct... should be given two groups

G_1, G_2

we can define

G_1 \times G_2

...
earlier on the page there seems to be an inconsistency, but I haven't done any work on Dihedral groups so I won't comment

Page 15, On the other hand... should be highest common factor of m and n not equal to one.

Spoiler

Reply 232

gasfxekl

Original post by Zacken

This works:

a, b \in \mathbb{Z} \Rightarrow (a+b) \in \mathbb{Z}

so closure, also

(a+b) + c = a + b + c = a + (b+c)

so associativity. Note

a + 0 = 0 + a = a

so identity and

a + (-a) = (-a) + a = 0

so inverse.

To be honest that's how you define Z, so this question seems a bit pointless.

Reply 233

Zacken

Original post by gasfxekl

So this question seems a bit pointless.

Agreed, I wouldn't have done it myself.

(edited 7 years ago)

Reply 234

liamm691

Original post by Zacken

If theres not much new content, am I missing out on much by doing Maths with Physics first year?

Reply 235

username1162972

Original post by liamm691

If theres not much new content, am I missing out on much by doing Maths with Physics first year?

Depends, do you already know fermats little theorem and all the elementary number theory ideas ?
The syllabus is online so you can look.

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Reply 236

EnglishMuon

For Q4 of exercise 1.3 (p11) of the Beardon book, is it necessary to look at the product of every element individually e.g. to show associativity/closure? I dont see that the cycles in different elements are disjoint so I dont think I can just conclude that

\alpha \beta = \beta \alpha

for alpha beta are any 2 elements in the set.

Reply 237

Zacken

Original post by EnglishMuon

\alpha \beta = \beta \alpha

for alpha beta are any 2 elements in the set.

Think this is a case where drawing a Cayley table is the best way to move forward. (google image a Cayley table if you don't know what it is, apologies if you do)

Reply 238

Insight314

Original post by liamm691

If theres not much new content, am I missing out on much by doing Maths with Physics first year?

You don't need any N&S to progress into theoretical physics, except maybe some elementary set theory and mathematical logic, but I would doubt the number theory knowledge would ever come up useful if you plan to specialise in theoretical physics. However, bare in mind that some DoS make you do N&S at the start in case you decide to drop Maths with Physics option since some Mathmos do choose to do that (sometimes due to practicals); I know for a fact Churchill DoS does that.

Original post by physicsmaths

Depends, do you already know fermats little theorem and all the elementary number theory ideas ?
The syllabus is online so you can look.
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Stop baiting fellow applied mathematicians into your purist ideologies!

Also, since you are such a 'know-it-all' with N&S, I have a question for you.

Take a look at Exercise 1.2 Question 6: "Let

\Omega

be a non-empty set and let

G

be the set of subsets of

\Omega

(note that

G

includes both the empty set

\emptyset

and

\Omega

. Why does it include the empty set? I Googled it and "powerset" came up on wikipedia. Is that some kind of thing you need to know from axiomatic elementary set theory?

Reply 239

EnglishMuon

Original post by Zacken

Think this is a case where drawing a Cayley table is the best way to move forward. (google image a Cayley table if you don't know what it is, apologies if you do)