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.
Best Answer
At least in ZFC, there is something called the axiom of extensionality which asserts that if $A$ and $B$ are sets with the same elements, then they are the same set, $A = B$.
In your example, both sets contains only three objects and exactly the same three objects $1, 2, 3$. Hence they are the same set so we may write $\{1,1,2,3\} = \{1, 2, 3\}$.