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A Summer of Maths (ASoM) 2016

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    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.
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    (Original post by 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.
    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)
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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?
    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?
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    (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)
    ah ok, thanks, yeah ive used a cayley table once or twice before Im guessing this is one of those checking you can do basic permutation manipulation questions then
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    (Original post by Insight314)
    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.



    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?
    Power set is the set of all the subsets of that set
    PS u may find Cantors theorem interesting
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    (Original post by EnglishMuon)
    ah ok, thanks, yeah ive used a cayley table once or twice before Im guessing this is one of those checking you can do basic permutation manipulation questions then
    Working through Beardon's, I see? How is it going?
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    (Original post by Insight314)
    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.



    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?
    Well he asked if hes missing out anything if N n S is easy, and that depends if you know the material or not, thats a pretty simple answer.
    I can't see what you wrote in latex on my phone plus im out so i havent got beardons book with me.
    Don't be jealous olympiaders are slick at number theory and already a course ahead.


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    (Original post by EnglishMuon)
    Power set is the set of all the subsets of that set
    PS u may find Cantors theorem interesting
    Yeah, I understand that but I don't get it why the empty set is included in all subsets. I just can't get the intuition behind it, and I don't want to move on without having any doubts unanswered.
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    (Original post by physicsmaths)
    Well he asked if hes missing out anything if N n S is easy, and that depends if you know the material or not, thats a pretty simple answer.
    I can't see what you wrote in latex on my phone plus im out so i havent got beardons book with me.
    Don't be jealous olympiaders are slick at number theory and already a course ahead.


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    I am not even doing N&S so not jealous at all, m8.
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    (Original post by Insight314)
    Working through Beardon's, I see? How is it going?
    ah good thanks! dyslexia means Im a very slow reader but I should find it a little faster wen on to a more familiar topic
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    (Original post by EnglishMuon)
    ah good thanks! dyslexia means Im a very slow reader but I should find it a little faster wen on to a more familiar topic
    Lol you are on page 11, I am on page 5 and I have worked through it for 3 hours now. Don't you take notes or am I more dyslexic than you, haha?
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    (Original post by Insight314)
    Yeah, I understand that but I don't get it why the empty set is included in all subsets. I just can't get the intuition behind it, and I don't want to move on without having any doubts unanswered.
    The empty set has no elements so all of its elements always are in any set.



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    (Original post by Insight314)
    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?

    For any set X we have (trivially) \emptyset \subset X. So the set of all subsets of X has to contain all the subsets of X, one of which is \emptyset.

    Think about what being a subset means. If I claim \emptyset \subset X, I'm saying that for all x \in \emptyset, I also have x \in X, this is a vacuously true statement.

    ETA: can you name me an element in \emptyset that is not in X?
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    (Original post by Insight314)
    Yeah, I understand that but I don't get it why the empty set is included in all subsets. I just can't get the intuition behind it, and I don't want to move on without having any doubts unanswered.
    Cus the empty set is just the set of 'nothing' i.e. no elements. Any set can contain 'nothing' as well as its other elements as in this case u should probably treat nothing as in something with no elements. e.g. we form subsets of a set by removing elements from the original set. If we remove every element, we are left with the null set.
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    (Original post by EnglishMuon)
    Cus the empty set is just the set of 'nothing' i.e. no elements. Any set can contain 'nothing' as well as its other elements as in this case u should probably treat nothing as in something with no elements. e.g. we form subsets of a set by removing elements from the original set. If we remove every element, we are left with the null set.
    So r u saying
    0=0 implies
    1+0=1
    Wowza


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    (Original post by Insight314)
    Lol you are on page 11, I am on page 5 and I have worked through it for 3 hours now. Don't you take notes or am I more dyslexic than you, haha?
    haha nah i dont take notes, havent wrote anything down for the past 3 years or so so thats probably why I also skipped most the basic GT stuff since ive covered that before.
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    (Original post by EnglishMuon)
    ah ok, thanks, yeah ive used a cayley table once or twice before Im guessing this is one of those checking you can do basic permutation manipulation questions then
    I think so, just checking your capacity for systematic thought, examining if you remember your definitions of a group, applying it to permutations and that, unless of course, I'm brutally wrong and there's a nice way to do it.
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    (Original post by EnglishMuon)
    haha nah i dont take notes, havent wrote anything down for the past 3 years or so so thats probably why I also skipped most the basic GT stuff since ive covered that before.
    How did u answer step questions then.
    U need answers to get marks u know.
    Cmon mayte.


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    (Original post by physicsmaths)
    So r u saying
    0=0 implies
    1+0=1
    Wowza


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    lol u know wat i mean

    Muons Theorem: Every number = 0
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    (Original post by physicsmaths)
    How did u answer step questions then.
    U need answers to get marks u know.
    Cmon mayte.


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    telekinesis to the examiner.
 
 
 
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