We use induction to prove that n horses are the same colour: Base case, n=1, true.
assume true for n=k \geq 1 If we have k+1 horses. Consider the first k horses. By assumption, they are all the same colour. Take one of these k horses, and the (k+1)th horse - again, by assumption, they are the same colour. Hence all k+1 are the same colour (here we use 'transitivity of colour' - to be formalised below)
------------------ Q2
Above i mentioned 'transitivity of colour'. Let S be a set, and ~ be a 'binary relation' on S. (this means that it represents true or false. It's like a magic ball, you can give it two elements a,b of S and ask it the question, are they related? and it will return true or false (This is notated as a~b) In maths this is often stated as ~:S×S→{true,false} where ′×′ denotes the Cartesian product (look it up!)
We say that '~' is: transitive if and only if (a~b and b~c implies a~c) for any a,b,c in S symmetric if and only if (a~b implies b~a) for any a,b in S reflexive if and only if (a~a) for any a in S
If all three of the above are satisfied, then ~ is called an equivalence relation.
Convince yourself that "having the same colour" is an equivalence relation on any set of elements with a colour. Question: Is ≥ is an equivalence relation? Is > an equivalence relation?
---------------- Q3 Let N⊂R be finite. Use the principle of induction to show that it has a least element.
------------------ Really tough question: The guests at Hilbert's hotel keep asking the receptionist if they can use the phone to make a call. The female receptionist at Hilbert's hotel wishes to keep track of how many times each room number has used the phone. She wishes to use her typewriter because he pencil broke and Mr Hilbert chewed her pen so that it no longer works (he was hungry). Again, Mr Hilbert has been silly and he chewed all but 2 of the keys off the receptionist's typewriter,
Q4 i) how can she keep her record?
Later that evening, after an unsatisfactory meal produced by Mrs Hilbert, Mr Hilbert is hungry again. Mr Hilbert eats another key from the typewriter, leaving just one key!
ii)It is possible for the receptionist to keep track of the room numbers, but how?
Spoiler
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The first few pages of chapter 8 in Korner's "A Companion to Analysis" are readable and somewhat surprising.
Typo, that k was meant to be a 1. This then gives f(k)=1 for all k.
Oh right! I couldn't see what you where getting at with the mistake but by having a go at it myself I can see it now! Hence I kind of unknowingly rephrased your 'corrected' solution above! Sorry!
What are people going to read up on over the summer?
I have put one of the books on which I'll spend some time over the summer. The topics that interest me are Group Theory, Analysis and some Vector Calculus.
At the Nottingham University Open Day there was a "Maths Trail" with several interesting questions on it. The questions ranged from requiring a working knowledge of arithmetic progressions and combinations to lowest common multiples and counting squares; and more emphasis was put on thinking about them than slogging through endless manipulation. I thought I'd share one with you. The question is of the kind that could be set in C1, but is an interesting one:
A rectangle is inscribed inside a circle of radius 6 units such that each of the vertices of the rectangle lie on the circumference of the circle. Given that the perimeter of the rectangle is 28 units, what is the area?
We randomly and independently generate elements of Ωrn times. If the sum of the results is rn+1=0 we stop. If the sum of the results is rn+1≥1, we randomly and independently generate elements of Ωrn+1 times etc.
Given that r1=1, what is the probability of this process ending?
yeah! I've been finding it interesting for some reason. I'll probably get bored of it soonish and turn back to pure though and gorge on Olympiad problems I imagine.
I have put one of the books on which I'll spend some time over the summer. The topics that interest me are Group Theory, Analysis and some Vector Calculus.
Spoiler
It looks good but it's a little on the expensive side given that this does a sufficiently good job, in my opinion (and is mentioned on one of the Colleges' online reading list, I forget which). If you want more stuff on Tensors, there's another book from the same series (although I haven't personally tried it); and you can buy both for a fraction of the price of Bourne's book.
At the Nottingham University Open Day there was a "Maths Trail" with several interesting questions on it. The questions ranged from requiring a working knowledge of arithmetic progressions and combinations to lowest common multiples and counting squares; and more emphasis was put on thinking about them than slogging through endless manipulation. I thought I'd share one with you. The question is of the kind that could be set in C1, but is an interesting one:
A rectangle is inscribed inside a circle of radius 6 units such that each of the vertices of the rectangle lie on the circumference of the circle. Given that the perimeter of the rectangle is 28 units, what is the area?
Required Knowledge:
Spoiler
Hints:
Spoiler
Full Solution:
Spoiler
Love these kind of problems Just a note though - could you not observe that you need the product of the roots of the quadratic equation, which is given by c/a in 0=ax^2+bx+c, so you know the answer is 26 without having to find the roots then multiply them together?