Zero of Power Set with Union

Theorem

Let $S$ be a set and let $\powerset S$ be its power set.

Consider the algebraic structure $\struct {\powerset S, \cup}$, where $\cup$ denotes set union.

Then $S$ serves as the zero element for $\struct {\powerset S, \cup}$.

Proof

We note that by Set is Subset of Itself, $S \subseteq S$ and so $S \in \powerset S$ from the definition of the power set.

From Union with Superset is Superset‎, we have:

$A \subseteq S \iff A \cup S = S = S \cup A$.

By definition of power set:

$A \subseteq S \iff A \in \powerset S$

So:

$\forall A \in \powerset S: A \cup S = S = S \cup A$

Thus we see that $S$ acts as the zero.

$\blacksquare$

Sources

• 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Exercise $1.4: \ 10$
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.3$. Units and zeros: Example $75$