Show that $A \bigcup B$ = $A \cap B$ implies $A = B$.

Step 1: Assume the given condition

We start by assuming the given condition that the union of sets $A$ and $B$ is equal to their intersection. Our goal is to prove that this condition implies that set $A$ must be equal to set $B$.

Step 2: Consider an element in A

To prove that $A = B$, we need to show that $A \subseteq B$ and $B \subseteq A$. Let's start by assuming an arbitrary element $x$ belongs to set $A$.

Step 3: Use the union property

If an element $x$ is in set $A$, then by the definition of union, $x$ must also be in the union of $A$ and $B$, denoted as $A \bigcup B$.

Step 4: Apply the given condition

From our initial assumption, we know that $A \bigcup B = A \cap B$. Therefore, if $x$ is in $A \bigcup B$, it must also be in $A \cap B$.

Step 5: Deduce A is a subset of B

If $x$ is in the intersection of $A$ and $B$, it means $x$ is in both $A$ and $B$. Since we started with $x \in A$ and concluded $x \in B$, this shows that every element of $A$ is also an element of $B$, meaning $A$ is a subset of $B$.

Step 6: Deduce B is a subset of A

By following the same logic, if we assume an arbitrary element $y$ is in $B$, then $y$ must be in $A \bigcup B$. Using the given condition, $y$ must also be in $A \cap B$, which implies $y$ is in $A$. Thus, $B$ is a subset of $A$.

Step 7: Conclusion

Since we have shown that $A$ is a subset of $B$ and $B$ is a subset of $A$, by the definition of set equality, it must be that $A = B$.

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