Zero Element of Union and Intersection

Source Work

 * Introduction to Set Theory
 * $1$. Elementary Operations on Sets
 * $1$. Elementary Operations on Sets

This mistake can be seen in the 2009 Dover edition: ISBN 0-486-47222-1

Mistake

 * "The operations $A \cup B$ and $A \cap B$ are meaningful for every pair of sets $A, B$ ... The empty set $\varnothing$ plays the role of the zero element."

This is technically incorrect.

From Zero of Power Set with Intersection, $\varnothing$ does indeed play the role of the zero element on the algebraic structure $\left({\mathcal P \left({S}\right), \cap}\right)$.

However, from Zero of Power Set with Union, the role of the zero element on the algebraic structure $\left({\mathcal P \left({S}\right), \cup}\right)$ is actually played by $S$.

In Gaal's defence, this book is not about abstract algebra but about topology.