SUMMARY
The equation $(A - B)\cup B = A$ holds true if and only if $B\subseteq A$. The proof begins with the assumption that $B$ is a subset of $A$, leading to the transformation of $(A - B)\cup B$ into $(A\cap B^c)\cup B$. This expression simplifies to $(A\cup B)\cap (B^c\cup B)$, which ultimately results in $A$. The confusion arises when interpreting the final result, where some participants mistakenly conclude that the expression equals $B$ instead of $A$.
PREREQUISITES
- Understanding of set theory concepts, including set difference and union.
- Familiarity with notation such as $A - B$, $B^c$, and subset relations.
- Basic knowledge of logical equivalences in mathematical proofs.
- Ability to manipulate and simplify set expressions using algebraic techniques.
NEXT STEPS
- Study the properties of set operations, particularly focusing on union and intersection.
- Learn about the universal set and its implications in set theory.
- Explore more complex proofs involving subsets and set identities.
- Review logical equivalences and their applications in mathematical reasoning.
USEFUL FOR
Mathematicians, students of set theory, and anyone interested in understanding the foundational principles of mathematical proofs involving sets.