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[SOLVED] finite sets
can someone please give me examples of finite sets as i don't really understand it.
So far as I know finite sets means the collection of elements such that the number of elements in the collection is 'countable'.
For example,
$\displaystyle A = \{1,2,3,4\}$ is a finite set because the number of elements in A is 4.
$\displaystyle B = \{a,b,c,d,e\}$ is also a finite set because you can count out the elements in B to be 5.
You may or may-not be familiar with the '...' (ellipsis)* notation where,
$\displaystyle A = \{1,2,...,4\}$ and $\displaystyle B = \{a,b,...,e\}$ both of these are finite sets mainly because - regardless of what the '...' represents - they do have a finial element, 4 being the last element in A and e being the last element in B.
*the ellipsis '...' represents all the elements that occur in a sequence up until the element after '...' appears in the sequence. Sometimes that sequence is obvious, sometimes not - it usually depends on the context of the set the '...' is used in, for example A could be a set of all natural numbers between 1 and 4 (inclusively) (which in the original example it is) - or, with the ellipsis in place, it could also mean all the natural numbers formed by the powers of 2 by which '...' would represent nothing at all.
However, if A and B were written as,
$\displaystyle C = \{1,2,...\}$ and $\displaystyle D = \{a,b,...\}$
then you can't deduce the number of elements in C and D - in other words, you can't count them up as you could before and so there is no 'finial' element and so C and D are examples of non-finite (infinite) sets.
Putting countable in quotes makes this wrong, as the term 'countable' includes denumerable (countably infinite) sets.Originally Posted by rain
Intuitively: a non-empty set $\displaystyle \text{A}$ is finite means that there exists a natural number $\displaystyle n$ such that there exists a one-to-one function between $\displaystyle \text{A}$ and the set $\displaystyle \{1,2, ..., n\}$
CB
No, wait, you've misunderstood me: I put 'countable' specifically in the quotes so as not to associated it with the other term; I was using it as an analogy because all finite sets (so far as I know) have a finite number of elements - and I thought this simple analogy would be helpful
You said you consider it wrong specifically because of the quote marks - whereas I would have considered it wrong without the quote marks; perhaps this is more of a cultural difference. For me, without the quotation marks implies this is a definition rather than an analogy
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Also, I'm quite curious to know what was changed in my original reply since I've writing it three days ago and can't remember it exactly
Edit:
That said, I suppose the quotation marks would in fact draw attention to the term; but now that has me wondering...
Because "countable" is a technical term which means something other than you want you should avoid it in this context and either use something else or make it explicit that you mean "can be places in a one-to-one correspondence with the set: {1, 2, ... n} for some n in the Naturals".
CB
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