Zero of Power Set with Intersection
Jump to navigation
Jump to search
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 $75$