Clarify the steps: what happened in this mathematical modelling of TSP?
You seem to have misunderstood pretty much every part of the statement

$$\sum_{i,j\in\{1,2,3\},i\neq j} x_{ij}=3&gt;2=|\{1,2,3\}|-1\,.$$


  I get that $x_{ij}$ is equal to 3, 


No, the sum of all values $x_{ij}$ where $i$ and&nbsp;$j$ are distinct values from $\{1,2,3\}$ is equal to&nbsp;$3$.


  but why the "> 2" ?


Because three is bigger than two.


  And what is the deal with subtracting 1 from a set? How do you even do that?


No, it's subtracting one from the cardinality of the set. Notice the $|\dots|$.


  How come $|\{1,2,3\}|-1 = 3 &gt; 2$   ?!?


It isn't. When we write something like $A=B&gt;C=D$, it means that $A=B$, $B&gt;C$ and $C=D$. You can't just re-order the terms and expect the statement to remain true, just as you can't reorder $3&gt;2$ as $2&lt;3$ and expect it to remain true.

So, the statement as a whole means:


The sum of the values $x_{ij}$ is equal to&nbsp;$3$.
Also, $3&gt;2$.
Also, $2=|\{1,2,3\}|-1$.



  So how is he allowed to write:
  $$|\{1,2,3\}|-1 = 3 &gt; 2\,?$$


He isn't and he doesn't.