$\{1,1\}=\{1\}$, origin of this convention Is there any book that explicitly contain the convention that a representation of the set that contain repeated element is the same as the one without repeated elements?
Like $\{1,1,2,3\} = \{1,2,3\}$.
I have looked over a few books and it didn't mention such thing. (Wikipedia has it, but it does not cite source).
In my years learning mathematics in both US and Hungary, this convention is known and applied. However recently I noticed some Chinese students claim they have never seen this before, and I don't remember I saw it in any book either. 
I never found a book explicitly says what are the rules in how $\{a_1,a_2,a_3,\ldots,a_n\}$ specify a set. Some people believe it can only specify a set if $a_i\neq a_j \Leftrightarrow i\neq j$. The convention shows that doesn't have to be satisfied. 

For variety, I'll note that both magma and python have a set constructor using a comma-separated list surrounded by curly braces, and they both allow repeated entries. For example, in python:
>>> {1,1,2,3}
{1, 2, 3}
>>> {3,2,1} == {1,1,2,3}
True

Mathematica, on the other hand, uses curly braces to construct lists rather than sets, so it would behave differently. Array initializers in C also use curly braces -- but again you're creating lists, not sets.
