Equivelence classes, how many there are, and how many elements they have. I've been struggling to understand equivalence classes. Say I have a set T, the set of all binary strings, and the relation S on T = {(a,b) | length(a) = length(b)}. 
How would I write down the equivalence class of 00? What about 101?
How do I know how many equivalence classes there are, or how many elements are in an equivalence class?
Thanks.
 A: Equivalence classes can be looked at in two different ways. One is in terms of the definition - for example the problem you have described above.
The other way is that an equivalence relation breaks a set into disjoint subsets (disjoint meaning they don't share any members). This is a much easier way of visualising an equivalence relation.
Consider all natural numbers N = {1, 2, 3 ....}. Define an equivalence relation R such that xRy if (x-y) is divisible by two. This means x and y are both even or both odd (try some numbers to see that this is true). So this equivalence relation divides all numbers into one of two disjoint sets, being {1,3,5,7 ...} and {2,4,6...}. If xRy either both x and y are both from the even set or both from the odd set.
Now lets look at your example. Possible strings are "0", "1", "00", "01", "10", "11", "000", "001" ...
The rule is that xRy only if they have the same number of digits (same length of string). So one equivalence class is all strings of length 1, ie {"1", "0"}. So for example "1"R"0" is a valid relationship.
Another equivalence class is all strings of length 2, being {"00", "01", "10", "11"). So for example "10"R"01" is true.
Another equivalence class is all strings of length 3, of which there are eight, {"000", "001" ..... "111"}.  "010"R"101" is true.
There is another equivalence class of size 16 (16 elements) for strings of length 4, and so on.
In the question, strings can be infinite, and these all form one single equivalence class - strings of infinite length. So for example "0000..."R"1111....", all infinite strings are in the same equivalence class.
When solving these, the easiest thing to do is defines the disjoint sets that form the equivalence classes - odd and even numbers in my first example, or sets comprising binary strings of length 1,2,3.. plus infinite strings. When you visualise these sets, its easy to see what the equivalence relation actually is.
Note that as your question seems to allow strings of zero length, there is one additional equivalence class which comprises all binary strings of length zero. Its only member is "" (the empty string) and the only relation is ""R"" so its the trivial case.
A: For an equivalence relation (one that is reflexive, symmetric, and transitive) an equivalence class of an element is the set of all elements which are in relation to that element.
For example, the equivalence class of $00$ can be written as $[00] = \{00,01,10,11\}$ since our relation is the "has the same length" relation, and each of $00,01,10,11$ have the same length (two).  Try to convince yourself why there are no other length two binary sequences that we missed.
So, the equivalence classes in this example are in direct correspondence to the lengths of the sequences.  There is an equivalence class with all 100-length sequences, and 31337-length sequences, etc.  How many are there?

 There are countably infinitely many equivalence classes, one for each element of $\mathbb{N}\cup\{\infty\}$  (as @PeterWebb pointed out in his answer, depending on your definitions, infinite length strings are often allowed as well and will comprise one of the equivalence classes)

Suppose that we are looking at the equivalence class of $\begin{matrix}\left[\underbrace{000\dots 0}\right]\\n~\text{copies}\end{matrix}$ (i.e. the equivalence class for the length-$n$ binary sequences).  How many different length-$n$ binary sequences are there?

 By multiplication principle, you have two choices for the first number (either 0 or 1), two choices for the second number, two choices for the third, ..., 2 choices for the $n^{th}$, for a combined total of $2^n$ different such sequences.

