Zero Element of Union and Intersection

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 $\O$ plays the role of the zero element.

This is technically incorrect.

From Zero of Power Set with Intersection, $\O$ does indeed play the role of the zero element on the algebraic structure $\struct {\powerset S, \cap}$.

However, from Zero of Power Set with Union, the role of the zero element on the algebraic structure $\struct {\powerset S, \cup}$ is actually played by $S$.

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

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets