Zero Element of Union and Intersection

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Source Work

1964: Steven A. Gaal: Point Set Topology:

Introduction to Set Theory
$1$. 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.


Sources