According to the principle of duality in set theory, if an identity involving union \( (\cup) \) and intersection \( (\cap) \) is true, then Select one: a. the same identity remains unchanged b. the identity is valid only for finite sets c. the identity remains valid when \( \cup \) and \( \cap \) are interchanged \( =\bigcirc \) d. the identity does not hold in general

Did you know that the principle of duality in set theory is a powerful tool that enhances not just mathematical rigor but also our understanding of relationships within sets? When you flip the symbols of union and intersection, you essentially unlock a parallel universe of identities! For example, De Morgan's laws beautifully demonstrate this duality. In real life, this principle plays a crucial role in fields like computer science, particularly in database query optimization. By understanding the duality of set operations, programmers can reframe problems and devise more efficient algorithms. So next time you're organizing your data or defining a query, think about how swapping union and intersection might simplify your task!