When doing a proof, is "considering" an arbitrary element equivalent to "fixing" an element? I don't understand the distinction, if any.
For example, here's a proof from Wikipedia that uses the term "fix."
So in the above proof, if I replace "fix" with "consider," is the proof logically still the same?If a monoid has the cancellation property and is finite, then it is in fact a group.
Proof: Fix an element x in the monoid. Since the monoid is finite, x^n = x^m for some m > n > 0. But then, by cancellation we have that x^(m-n) = e where e is the identity. Therefore x * x^(m-n-1) = e, so x has an inverse.
Sometimes people even say, "Consider an arbitrary but fixed element...""fixing an element" is generally used when one is going to use a "FIXED" element in several instance and we want to be sure it is always the same FIXED element and not other one.