If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is A ∩ B?
A{1, 2, 3, 4, 5, 6}
B{3, 4}
C{1, 2, 5, 6}
D{}
A ∩ B contains elements in both A and B simultaneously. Only 3 and 4 appear in both sets. Option A is A ∪ B (elements in either set). Option C is the symmetric difference (elements in one but not both). Option D would be the case if the sets shared no elements.
Question 2 True / False
For any sets A and B, the union A ∪ B usually contains strictly more elements than the intersection A ∩ B.
TTrue
FFalse
Answer: False
If A = B, then A ∪ B = A ∩ B = A — they are identical and have the same number of elements. More generally, A ∪ B = A ∩ B if and only if A = B. The union is always at least as large as the intersection, but equality is possible.
Question 3 Short Answer
Describe in words what the set (A ∪ B)^c represents, relative to a universal set U. Then name the equivalent expression given by De Morgan's law.
Think about your answer, then reveal below.
Model answer: It is the set of all elements in U that belong to neither A nor B. By De Morgan's law, (A ∪ B)^c = A^c ∩ B^c.
De Morgan's laws link complement with union and intersection: the complement of a union is the intersection of the complements, and vice versa. Intuitively, to be outside (A ∪ B) means to be outside A AND outside B simultaneously — which is exactly A^c ∩ B^c. These laws are frequently used to simplify set expressions and to rewrite logical conditions.