Solution

Relation $R$ is defined as:

$(a,b)\in R$ if $a$ is a brother of $b$.

Symmetric:

If $a$ is brother of $b$, then it is not necessary that $b$ is brother of $a$.

For example, $b$ can be female. So, $b$ will not be brother of $a$.

Therefore, $R$ is not symmetric.

Transitive:

If $a$ is brother of $b$ and $b$ is brother of $c$, then $a$ is also brother of $c$.

Therefore, $R$ is transitive.

Equivalence relation:

For equivalence relation, relation must be reflexive, symmetric and transitive.

Here, $R$ is not symmetric and not reflexive.

So, $R$ is not an equivalence relation.

Partial order relation:

For partial order relation, relation must be reflexive, antisymmetric and transitive.

Here, $R$ is not reflexive.

So, $R$ is not a partial order relation.