Here's a homework question I'm probably over thinking.
Prove that if $S$ is a set and $A$ is a nonempty proper subset of $S$, then $\Pi = \{A,S\setminus A\} $ is a partition of $S$.
The claim is clearly true, however I'm having difficulty structuring my proof. I believe that if $\{A, S\setminus A\}$ is partition of $S$, then $A \cup S\setminus A = S$ and $A \cap S\setminus A = \emptyset$.
Is showing these conditions are true all that required to prove the statement above, or is there something I'm missing? Thank you.
After looking back at the definition used in our textbook,
Let $S$ be a nonempty set. A partition $\Pi$ of $S$ is a family of $\Pi = \{A_i\}_{i\in I}$ of nonempty subsets of $S$ satisfying these conditions:
1) $\bigcup_{i\in I}A_i = S$,
2) $A_i \cap A_j = \emptyset$ if $i \neq j$.
It's clear to me now that to prove something is a partition of a set, you can simply prove these two conditions are true.