How can I show that this has a reflexive closure? Assume R is a relation on A.
Show that R ∪ ΔA is the reflexive closure of R.
What I thought is, that if R is an union with ΔA, it must mean that all a in A are (a,a).
But I don't know if this is good argument.

The reflexive closure of $R$ is the smallest reflexive relation that contains $R$.  So your argument needs three parts:


*

*$R\cup\Delta A$ contains $R$

*$R\cup\Delta A$ is reflexive

*If $S$ is a proper subset of $R\cup\Delta A$, then it fails (1) or (2).  


Point 1 is trivial (it follows from basic properties of set union).  But you need to establish 2 and 3.
Item 3 states that $R\cup\Delta A$ is the smallest relation that satisfies (1) and (2), because it says that if $S$ is smaller than $R\cup\Delta A$, then it does not satisfy (1) and (2).  This is a crucial point.
The argument you stated in your post (“all $a$ in $A$ are $(a,a)$”) does not make sense to me.
