If and what is (A∪B)∩(A∪C)?
If A={1,2,3}, B={3,4,5}, and C={1,2,6}, what is (A∪B)∩(A∪C)?
- A.
{1,2,3}
- B.
{1,2,3,4}
- C.
{1,2}
- D.
{3}
Attempted by 30 students.
Show answer & explanation
Correct answer: A
For any sets A, B, C, intersection distributes over union: (A∪B)∩(A∪C)=A∪(B∩C). This identity means you can either compute the two unions directly and intersect them, or compute B∩C first and union it with A — both routes must agree.
Compute A∪B: combine every element of A={1,2,3} and B={3,4,5}, giving A∪B={1,2,3,4,5}.
Compute A∪C: combine every element of A={1,2,3} and C={1,2,6}, giving A∪C={1,2,3,6}.
Intersect the two unions: keep only elements present in both {1,2,3,4,5} and {1,2,3,6}, giving (A∪B)∩(A∪C)={1,2,3}.
Cross-check using the distributive identity: B∩C={3,4,5}∩{1,2,6}=∅ (B and C share no elements), so A∪(B∩C)=A∪∅=A={1,2,3} — the same result confirms the answer.