“Isn’t that cute?”

@dt  
It’s true that in a set with just one element then 0 = 1 (trivial), but the contrary is true only in a ring. In short, if a(b + c) = ab + ac.
 
Rings are what people naturally think about when dealing with addition and multiplication.

@BigMax  
If the elements are the zero
 
n=n+0=n+1 -> all the elements in the set are n  
n=n1=n0=0 -> all the elements in the set are 0
 
Unless n0 != 0.
 
Maybe? I haven’t done math since college.

@dt  
Well, in Mathematics “equal” is always relative to a symmetrical, reflexive and transitive relation, so we have something to start with. Nothing prevents you to change the definition in your context, but that would be a mess.
 
But “0” and “1” aren’t always what people used to think they are, i.e. the first natural number and its successor. If you define two binary operations in a set of elements, and call them “addition” and “multiplication”, then “0” and “1” can be defined as the neutral elements of those operations, respectively. If the elements are the same, then 0 = 1.

@BigMax
 
Or equal.

@Gundlach  
It depends on how you define 0 and 1.

“One can never equal zero!”

Only if you don’t understand it, it doesn’t mean it’s wrong.  
I thought you learnt that lesson.

@Princess♠Molestia  
Objection overruled.