CLEP College Mathematics: Set Operations Practice Questions
Work with finite sets, complements and overlapping groups in eight original questions. Pay attention to the stated universal set and to whether a question asks for either group, both groups or neither group.
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Answer the questions below — you get instant feedback and a full explanation for each.
1. Let A = {1, 3, 5} and B = {3, 4, 5}. What is A union B?
Explanation. The union contains each element that belongs to either set, listed once: 1, 3, 4 and 5. The intersection would contain only 3 and 5. Repeating an element does not create a new set member.
2. Within U = {1, 2, 3, 4, 5, 6, 7, 8}, let A be the even numbers and B the numbers greater than 5. What is A intersection B?
Explanation. Intersection requires both conditions. The numbers greater than 5 are 6, 7 and 8; only 6 and 8 are also even.
3. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {3, 6, 9}. What is the complement of A relative to U?
Explanation. Remove the members of A from U. The remaining seven members form the complement. Zero is excluded because it is not in U; 10 must remain because it is in U but not in A.
4. If A = {p, q, r, s} and B = {q, s, t}, what is the set difference A minus B?
Explanation. A minus B keeps elements in A that are not in B. Removing q and s from A leaves p and r. The element t is not included because it never belonged to A.
5. In a group of 64 people, 37 read magazine A, 29 read magazine B and 12 read both. How many read neither magazine?
Explanation. The number reading at least one is 37 + 29 - 12 = 54; subtract the overlap once because it was counted twice. Thus 64 - 54 = 10 read neither.
6. How many subsets does the set {u, v, w} have, including the empty set and the set itself?
Explanation. For each of three elements there are two independent choices: include it or leave it out. That gives 2 times 2 times 2 = 8 subsets. This count includes the empty set and the full set.
7. Complements are taken within the same universal set U. Which set equals the complement of A union B?
Explanation. An element outside A union B is in neither A nor B. It therefore lies in both complements. This gives the set form of De Morgan's law: complement(A union B) = complement(A) intersection complement(B).
8. Let P = {-2, 0, 2} and Q = {1, 3, 5}. Which statement is true?
Explanation. The sets share no elements, so their intersection is the empty set and they are disjoint. Zero belongs to P alone, so it cannot belong to the intersection.
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FAQ
How do union and intersection differ?
Union accepts membership in either set, including both. Intersection requires membership in both.
Why must a complement specify a universal set?
A complement consists of eligible elements outside the chosen set. The universal set specifies which elements are eligible.
How can I check an overlapping-group calculation?
Split the population into A only, B only, both and neither. These four counts must add to the total, and none can be negative.