$F, G$ partitions of A. show $F*G$ = ${\{t | t\in X, t\in Y\}}$ - where $X\in F$, and $Y \in G$

Question

$F, G$ are partitions of the group A. How can one show that $F*G$ is also a partition of $A$ in a way that $$F*G = \{t | t\in X, t\in Y\}$$ where $X\in F$, and $Y \in G$

Answer 1

Show it by hand:

1. Show that every point of $A$ appears in a set of $F * G$.
2. Show that no point of $A$ appears in more than one set of $F * G$.

First, you need to understand the definition of $F*G$, maybe write it down correctly (what you gave above is something like the idea, not the formal definition).