For all sets $A, B$, and $C$, prove that $(A \cap B) \cup C=(A \cup C) \cap(B \cup C)$. Cite a property from Theorem 6, 2.2 for every step.

Question

Anonymous · Answer

We proved the identity ( A ∩ B ) ∪ C = ( A ∪ C ) ∩ ( B ∪ C ) by applying the distributive property of sets. This property states that unions and intersections can be rearranged without changing the overall result. Thus, these two expressions represent the same set. 
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GinnyAnswer · Answer

Apply the distributive property of union over intersection: ( A c a pB ) c u pC = ( A c u pC ) c a p ( B c u pC ) . 
 The left-hand side (LHS) is ( A c a pB ) c u pC . 
 The right-hand side (RHS) is ( A c u pC ) c a p ( B c u pC ) . 
 Since LHS = RHS, the identity is proven: ( A c a pB ) c u pC = ( A c u pC ) c a p ( B c u pC ) ​ . 
 
 Explanation 
 
 Understanding the Problem We are given the task of proving the set identity ( A c a pB ) c u pC = ( A c u pC ) c a p ( B c u pC ) using properties from Theorem 6, 2.2. This identity is known as the distributive property of union over intersection. 
 
 Setting up the Proof Starting with the left-hand side (LHS) of the equation, we have ( A c a pB ) c u pC . We want to show that this is equal to the right-hand side (RHS), which is ( A c u pC ) c a p ( B c u pC ) . 
 
 Applying the Distributive Property We will use the distributive property of union over intersection, which states that for any sets A , B , and C , ( A c a pB ) c u pC = ( A c u pC ) c a p ( B c u pC ) . This property is exactly what we need to prove the given identity. 
 
 Conclusion Thus, we have shown that ( A c a pB ) c u pC = ( A c u pC ) c a p ( B c u pC ) . The left-hand side is equal to the right-hand side, so the identity is proven.

Examples 
 The distributive property of sets is used in computer science, particularly in database management and data mining. For example, when querying a database, you might want to find all customers who have either bought both product A and product B, or are located in city C. This can be represented using set operations, and the distributive property helps simplify and optimize such queries.

For all sets $A, B$, and $C$, prove that $(A \cap B) \cup C=(A \cup C) \cap(B \cup C)$. Cite a property from Theorem 6, 2.2 for every step.

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