A Summer of Maths (ASoM) 2016

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?

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

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

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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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

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.

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?

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.

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?

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

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.

So r u saying
0=0 implies
1+0=1
Wowza


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How did u answer step questions then.
U need answers to get marks u know.
Cmon mayte.


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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 $X$?

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.

Are you referring to the composition of permutations in a symmetry group?

If so, the fact that disjoint cycles commute should intuitively be obvious. Disjoint cycles operate on different numbers.

Formally,

Take disjoint cycles a,b. The permutation ab 'does b first, then does a', and the permutation ba does this in the opposite order. We can show that ab = ba by showing that each element has the same image for ab and for ba.

Suppose an element x is in cycle b, and b takes x to y. Then y is in b. So y is not in a (a does not 'act' on y) and ab takes x to y. Now suppose u is in cycle a, taken to v. Since u is not in cycle b, overall ab will take u to v.

Similarly you can show that ba will also take x to y, and u to v. Therefore ab = ba. They are the same function (permutation).


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