Zero 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 the empty set $\O$ serves as the zero element for $\struct {\powerset S, \cap}$.


Proof

From Empty Set is Element of Power Set:

$\O \in \powerset S$

From Intersection with Empty Set:

$\forall A \subseteq S: A \cap \O = \O = \O \cap A$

By definition of power set:

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

So:

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

Thus we see that $\O$ acts as the zero element for $\struct {\powerset S, \cap}$.

$\blacksquare$


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 $75$
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