Final Answer:
(a) A ∪ B = A ∪ B is always true.
(b) Yes, the result of closures in part (a) extends to infinite unions of sets.
Step-by-step explanation:
In part (a), we aim to prove that A ∪ B is equal to A ∪ B. By the definition of union, A ∪ B contains all elements that are in A or in B (or in both). When comparing A ∪ B to itself, we see that both expressions represent the same set of elements, as they use the same symbols A and B. Therefore, A ∪ B = A ∪ B is true by the reflexive property of equality.
For part (b), the result extends to infinite unions of sets. In set theory, the union of an infinite collection of sets is defined as the set of all elements that belong to at least one set in the collection. Since A ∪ B = A ∪ B holds for any finite sets A and B, it also holds when considering an infinite collection of sets. The property persists because the union operation is associative and commutative, allowing the extension to infinite unions without affecting equality.
In summary, the proof in part (a) relies on the reflexive property of equality, establishing that A ∪ B is indeed equal to A ∪ B. In part (b), the extension to infinite unions is justified by the nature of the union operation and its compatibility with the equality relation.