# Identity of Power Set with Intersection

## Theorem

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

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

Then $S$ serves as the identity for $\struct {\powerset S, \cap}$.

## 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 Intersection with Subset is Subset, we have:

- $A \subseteq S \iff A \cap S = A = S \cap A$

By definition of power set:

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

So:

- $\forall A \in \powerset S: A \cap S = A = S \cap A$

Thus we see that $S$ acts as the identity element.

$\blacksquare$

## Sources

- 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): Exercise $1.4: \ 9$ - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.3$. Units and zeros: Example $75$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 4$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Operations