Meaning of counting argument? 
*

*Does "counting argument" mean a proof of  some statements by
counting something?

*Is "counting argument" same as "double counting"?  Or does it include both double counting and  bijective proof?


I found some relevant Wikipedia articles Combinatorial  proof and Combinatorial principles. But not sure the meaning of counting argument.
Thanks!
 A: Counting argument isn’t really a well-defined term, but I’d understand it to mean a proof that two expressions are equal by showing that they are two different ways of counting the same thing. Wikipedia calls this double counting, which I consider a very poorly chosen name: to me double counting refers to counting some things twice and therefore having to correct the total by subtracting the amount of over-counting. I prefer to call it simply counting the same thing in two different ways.
However, I can imagine someone using the term counting argument to include bijective proofs as well as proofs by counting the same thing in two ways.
A: Counting argument can mean counting the same thing in two different ways to give rise to two different expressions that then have to be equal. For example, we have $\binom{n}{k}=\binom{n}{n-k}$.
Double counting can mean counting the same object twice so that one has to subtract to get the correct number. For example, we have $|A\cup B|=|A|+|B|-|A\cap B|$.
A: Tim,
Remember, any set of finite size is trivially countable.
Typically a "counting argument" refers to listing elements of a set in a meaningful way to show that the set is the the same size as the natural numbers or not the same size as the natural numbers.
