Asymmetric Relation Confusion

Question

$A = \{1, 2, 3, 4\}, R \subset A \times A$

Why is $\{(1,1),(2,2),(3,3)\}$ an asymmetric relation?

$(a,b)$ where $a=b$ must come under symmetric relation.

Answer 1

The relation you give is in fact not actually an asymmetric relation. The definition of an asymmetric relation is that if $(a,b)\in R$, then $(b,a)\not\in R$. This does not hold because $(a,a)\in R$.

On the other hand, the given relation is an antisymmetric relation. The definition of antisymmetric is that if $(a,b)\in R$ and $(b,a)\in R$, then $a=b$. Notice that our set satisfies this definition. The definition may not be completely clear, and the way I like to think of it is that if one of $(a,b)$ or $(b,a)$ is in the set (for $a\ne b$ !) then the other isn't.

It is important to note that neither asymmetric nor antisymmetric is the opposite of symmetric - indeed the relation you give is both symmetric and antisymmetric, and we can look at the relation $\{(a,a),(a,b)\}$ to see that a relation may not be symmetric and not be asymmetric at the same time.