Let the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}. What is the set (A ∪ B) ∩ (A ∩ B)^c?
Show the answer and explanation
Correct answer
A. {1, 2, 3, 6, 7}
Principle or equation
The complement of a set is with respect to the universal set. The union combines elements, the intersection takes common elements, and the complement takes elements not in the set.
Why this answer is correct
A ∪ B = {1,2,3,4,5,6,7}. A ∩ B = {4,5}. The complement of A ∩ B is U minus {4,5} = {1,2,3,6,7,8,9,10}. Intersecting A ∪ B with this complement gives {1,2,3,6,7}.
Example
If U = {1,2,3}, A = {1}, B = {2}, then A ∪ B = {1,2}, A ∩ B = ∅, complement of ∅ is U, so (A ∪ B) ∩ U = {1,2}.
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