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 $\O$ 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