If A = {1, 2, 3} and B = {2, 3, 4, 5}, what is A ∩ B?
A{1, 2, 3, 4, 5}
B{2, 3}
C{1, 4, 5}
D{}
A ∩ B contains elements that are in BOTH A and B. Checking each element: 1 is in A but not B (excluded). 2 is in both (included). 3 is in both (included). 4 is in B but not A (excluded). 5 is in B but not A (excluded). So A ∩ B = {2, 3}. Option A is the union, not intersection. Option C contains elements in exactly one set (the symmetric difference).
Question 2 True / False
If A and B are disjoint sets, then A ∪ B = ∅.
TTrue
FFalse
Answer: False
Disjoint means A ∩ B = ∅ (they share no elements), not that their union is empty. A = {1, 2} and B = {3, 4} are disjoint, but A ∪ B = {1, 2, 3, 4}. The union of disjoint sets combines all elements from both sets. Only if both A and B are themselves empty would A ∪ B be empty.
Question 3 Short Answer
A class has 30 students. 18 play basketball, 14 play soccer, and 7 play both. How many play at least one sport? Explain using union and intersection.
Think about your answer, then reveal below.
Model answer: Let B = basketball players, S = soccer players. |B ∪ S| = |B| + |S| - |B ∩ S| = 18 + 14 - 7 = 25. So 25 students play at least one sport.
The union B ∪ S represents students who play basketball OR soccer (or both). Simply adding 18 + 14 = 32 overcounts the 7 students who play both, because they are counted once in |B| and once in |S|. Subtracting the intersection corrects the overcount. This is the inclusion-exclusion principle.