$\{1,1\}=\{1\}$, origin of this convention

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 $\{1,1,2,3\} = \{1, 2, 3\}$.

It all ties back into how this specification of sets are defined.

An unordered tuple $\{a_1,a_2,a_3,a_4\dots\}$ is defined as $\{x:x=a_1 \lor x=a_2 \lor x=a_3 \lor x=a_4 \lor\dots\}$.

So, by this convention, $\{1,1\}$ = $\{x:x=1 \lor x=1 \}$

This is equal to $\{ x : x = 1 \}$ by the idempotency of $\lor$, so

$\{1,1\} = \{1\}$

I took a quick look through some of the likelier candidates on my shelves. The following introductory discrete math texts all explicitly point out, with at least one example, that neither the order of listing nor the number of times an element is listed makes any difference to the identity of a set:

• Winfried K. Grassman & Jean-Paul Tremblay, Logic and Discrete Mathematics: A Computer Science Perspective
• Ralph P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction, 4th ed.
• Richard Johnsonbaugh, Discrete Mathematics, 4th ed.
• Bernard Kolman, Robert C. Busby, & Sharon Ross, Discrete Mathematical Structures for Computer Science, 3rd ed.
• Edward Scheinerman, Mathematics: A Discrete Introduction, 2nd ed.