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A Summer of Maths (ASoM) 2016

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    (Original post by Insight314)
    What do you mean? I was bored before it arrived, but now gonna work through the textbook.
    Lol it was another joke haha.


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    (Original post by Insight314)
    Why not do both at the same time? That is why I am studying from Beardon. Physicsmaths has got the book and yet can't just work through it lol.
    Don't really care tbh.
    I have the book.
    V n M most of it I know anyway.


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    (Original post by Insight314)
    Beardon is like 300 pages and teaches you V&M and Groups at the same time whilst if you were reading through the LNs you would need to get through V&M first and then Groups, whereas for people who want to do Groups first, the first chapter of Beardon's textbook introduces them to Groups. It seems like the lazy person would choose Beardon's over going through lecture notes for V&M and Groups separately.
    Just do what ur doing. No ones gna change their mind after u tell them.


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    (Original post by physicsmaths)
    Just do what ur doing. No ones gna change their mind after u tell them.


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    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 in G such that for all g in 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).
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    (Original post by Insight314)
    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.
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    (Original post by Zacken)
    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.
    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.
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    (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 in G such that for all g in 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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    (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 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.
    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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    (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 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.
    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.
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    (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.
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    (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.
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    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).
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    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?
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    (Original post by 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?
    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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    (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.
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    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:
    Show
    In before 'but why?' - I need something to do.
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    (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.
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    (Original post by gasfxekl)
    So this question seems a bit pointless.
    Agreed, I wouldn't have done it myself.
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    (Original post by Zacken)
    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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    If theres not much new content, am I missing out on much by doing Maths with Physics first year?
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    (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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