TITLE: $\{1,1\}=\{1\}$, origin of this convention
QUESTION [12 upvotes]: 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.

REPLY [3 votes]: You asked

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?

Set Theory and the Continuum Problem by Smullyan and Fitting contains, on page 19:

For any sets $a$ and $b$ (whether the same or different) by $\{a,b\}$ we mean the class whose only elements are $a$ and $b$—or equivalently the class of all $x$ such that $x=a$ or $x=b$. … Note that if $a$ and $b$ happen to be the same, then $\{a,b\} = \{a\}$—stated otherwise, $\{a,a\}=\{a\}$.


Here is one of many examples that I found by searching in Google Books for set theory ordered pair. It appears on page 23 of Naive Set Theory by Paul Halmos.  This well-known book says:

The ordered pair of a and b… is the set $(a, b)$ defined by:
  $$(a, b) = \{\{a\}, \{a,b\}\}.$$
  …
  We note first that if $a$ and $b$ happen to be equal, then the ordered pair
  $(a, b)$ is the same as the singleton $\{\{a\}\}$.