In the topology,Let $X=\{1,2,3\}$ and let $d:X \times X \to[0,+ ∞)$ be a function defined by formulas

Question

Let $X=\{1,2,3\}$ and let $d:X \times X \to[0,+\infty)$ be a function defined by formulas: $$ d(1,1)=d(2,2)=d(3,3)=0,\\ d(1,2)=d(2,1)=d(1,3)=d(3,1)=1,\\ d(2,3)=d(3,2)=3.$$

Check if $d$ is a distance function on $X$.

Answer 1

To check that $d$ is a distance on $X$ you need:
1) $d(x,y)\geq 0$ $\forall x,y \in X$ and $d(x,y)=0$ iff $x=y$
2) $d(x,y)=d(y,x)$ $\forall x,y \in X$
3) $d(x,y) \leq d(x,z)+d(z,y)$ $\forall x,y,z \in X$

The first line of formulas gives you 1), the second line gives you 2), but then
$3=d(2,3) \leq d(2,1)+d(1,3)=1+1=2$ and that is a contradiction. Therefore $d$ is not a distance