Describe A ∩ (B ∪ C) for multiples of 2, 5, and 10 in N: Let A = {x ∈ N : x is a multiple of 2}, B = {x ∈ N : x is a multiple of 5}, and C = {x ∈ N : x is a multiple of 10}. Describe A ∩ (B ∪ C).

Introduction / Context:
Set expressions with arithmetic properties often reduce via containment. Because every multiple of 10 is a multiple of 5, C ⊂ B. This simplifies unions and intersections substantially.

Given Data / Assumptions:

• A = multiples of 2
• B = multiples of 5
• C = multiples of 10
• All subsets considered within N

Concept / Approach:
First, simplify B ∪ C. Since C ⊂ B, B ∪ C = B. Then A ∩ (B ∪ C) = A ∩ B = numbers divisible by both 2 and 5, i.e., multiples of lcm(2,5) = 10, which is precisely C.

Step-by-Step Solution:
C ⊂ B ⇒ B ∪ C = BA ∩ B = multiples of 10Hence A ∩ (B ∪ C) = C

Verification / Alternative check:
Pick examples: 10, 20, 30 are in both A and B, matching C exactly.

Why Other Options Are Wrong:
A and B are too large; “None of these” is unnecessary since C fits perfectly.

Common Pitfalls:
Forgetting that B ∪ C collapses to B due to subset containment; always simplify unions first.

Final Answer:
C