Why is the following relation anti-symmetric?
{(1,2), (2,3), (3,4), (1,4), (1,3), (2,4)}
From my understanding, it is anti-symmetric if:
$$ (a, b) \in R, (b, a) \in R, b=a $$
Must hold. Which is it this case not true. I would rather say it is asymmetric since there is no symmetric nor anti symmetric relation at all.
The relation is antisymmetric because the implication in the definition (which you have omitted) is vacuously true.
The relation is anti symmetric. There exist no pairs $(a,b)$ such that $(a,b)\in R$ and $(b,a)\in R$. This means that for every such pair, $a=b$ is true. It is also true that for all such pairs, the pope is an elephant from Mars.
First of all, I think you're misunderstanding the definition of an antisymmetric relation:
You write $$(a, b) \in R, (b, a) \in R, b=a$$ as though all three conditions separated by commas must hold.
The correct definition is as follows: For all $a, b$ in the given set on which a relation is defined,
$$\text{IF}\;(a, b) \in R \,\text{ AND }\, (b, a) \in R, \;\text{ THEN }\, a= b$$
If there are no cases where $(a, b)\in R$ and $(b, a) \in R$, then the relation is vacuously antisymmetric.
Another way to think about antisymmetry is as follows: A relation fails to be antisymmetric if and only if there exist $a, b$ such that $(a, b) \in R$ and $(b, a) \in R$, AND $a\neq b$
