Can you spot the mistake?
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)
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 ~ 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.
Is is an equivalence relation? Is > an equivalence relation?
Let 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?
ii) there are infinitely many primes.
any natural number has a unique prime factorisation. (Fundamental Theorem of Arithmetic)
The first few pages of chapter 8 in Korner's "A Companion to Analysis" are readable and somewhat surprising.
it is found here: ftp://22.214.171.124/books/DVD-002/Ko...on_to_Analysis[c]_A_Second_First_and_First_Second _Course_in_Analysis_(2004)(en)(6 13s).pdf
Last edited by jj193; 01-07-2012 at 22:55.
Does this work?
(Original post by jack.hadamard)
such that for all
explicitly specifies the set of all natural numbers together with zero; i.e. the union of the natural numbers with the singleton that contains zero.
Last edited by Lord of the Flies; 01-07-2012 at 23:19.
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!
(Original post by james22)
Typo, that k was meant to be a 1. This then gives f(k)=1 for all k.
Last edited by Lord of the Flies; 01-07-2012 at 23:09.
i) Let be an arithmetic sequence with non-zero terms.
Obtain an expression for .
ii) Find the sum to infinity of the following series.
Last edited by jack.hadamard; 02-07-2012 at 12:34.
Reason: Forgot about something. :p