Relation that is not an equivalence

Question

I was asked to show that the relation on the set $A=\{a,b,c,d\}$ definde by

$R=I_A\cup\{(a,b),(b,a),(c,d),(d,a)\}$

is an equivalence relation (where $I_A=\{(x,x)|x\in A\}$). It is obvious to me that this relation is niether symmetric nor transitive, so this is not an equivalence relation. Am I right?

Answer 1

Yes you are right.
By definition of symmetry, as you have $(c,d)$ in $R$, you should have $(d,c)$ in $R$. But you don't have that.
Again by definition of transitivity, as you have $(c,d)$ and $(d,a)$ in $R$, you should have $(c,a)$ in $R$. But you don't have that.
So $R$ is not equivalent.