TITLE: Negation of $0 = 1$
QUESTION [12 upvotes]: I'm taking my first proof-heavy class (real analysis), and one practice problem on the first homework is to write the negation of 
$$0 = 1$$
My immediate thought was that it would simply be
$$0 \neq 1$$
but I'm not 100% certain of that answer. I was wondering if there's more to it than just inverting the $=$ sign, and perhaps you'd distribute the negation like
$$\neg 0 \neq \neg1$$
but logically that doesn't make sense to me. I've tried looking this up, but a statement as simple as $0 = 1$ has given me a hard time finding any good search results.
Basically to break down my questions:

Is $0 \neq 1$ right?


if so, do I prove it somehow?
if not, how do you negate expressions like $\langle expr \rangle = \langle expr \rangle$?

REPLY [6 votes]: It depends a bit on your class. Were the natural numbers defined as sets? In Zermelo-Fraenkel 0 would be the empty set, and $1$ the set containing the empty set. So $0=1$ can be written as: for all $x$ in $1$ : $x \neq x$
Negation would be: there exists an x in 1 : $ x=x$.
It depends on the definitions used in class. Although if your class didn't introduce the numbers, probably the obvious answer $ 0\neq 1$ is requested.
Edit: an introduction to the set theoretic construction of the natural numbers can be found in the highest rated answer here: Set theoretic construction of the natural numbers