For the union of sets we use the union operator $ \cup$ (or $ \bigcup$ ). And for a concatenation (Kleene star) we also use the union operator. The operations are different, but why the same terminology and operator?

The following is my understanding of the union of sets versus the concatenation of sets (Kleene star). Please correct me if I’m wrong.

**Union of sets**

For the two sets $ \{a,b\}$ and $ \{a,b\}$ we have the union \begin{align} \{a,b\}\cup\{a,b\}=\{a,b\} \end{align}

**Concatenation of sets (Kleene star)**

The concatenation of $ \{a,b\}$ and $ \{a,b\}$ is also a union *(same notation?!)* of two sets \begin{align} \{a,b\}^*&=\bigcup _{i=0}^{2} \{a,b\} ^2=\{a,b\} \cup \{a,b\}\ &=\{\epsilon,a,b,aa,ab,ba,bb,aaa,aab,aba,abb,baa,\dots\} \end{align}