GRADE 11 · A4 PRINT VIEW
Sets · Answer guide
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Lesson 1 · Describe membership precisely
Represent a finite set and distinguish membership from inclusion.
How many elements are in {2,2,5,7}?
Answer: 3. Repetitions are ignored: the distinct elements are 2,5,7.
If A={2,4,6}, which is true?
Answer: 4∈A. 4 is listed as an element of A.
Which is a subset of {a,b,c}?
Answer: {a,c}. Both a and c belong to the larger set.
Discuss: Find a vague collection, rewrite its rule to remove opinion, and list its members.
Teaching note: The number 2 and the set {2} are different objects. Use membership for an element and inclusion for a set. Open responses need discussion; they are not automatically marked.
Lesson 2 · Combine sets without double counting
Calculate union, intersection and the size of a union.
If A={1,2}, B={2,3}, what is A∪B?
Answer: {1,2,3}. Union includes each element appearing in either set.
For the same sets, A∩B is…
Answer: {2}. 2 is the only shared member.
If |A|=7, |B|=6 and |A∩B|=2, find |A∪B|.
Answer: 11. 7+6−2=11.
Discuss: Create two sets with four members each whose union has six members. Explain the overlap.
Teaching note: Union does not mean add the cardinalities without checking the overlap. Open responses need discussion; they are not automatically marked.
Lesson 3 · Use the universe when taking a complement
Explain why a complement depends on the chosen universal set.
If U={1,2,3,4} and A={2,4}, A′ equals…
Answer: {1,3}. 1 and 3 are in U but not A.
A∩A′ is always…
Answer: Empty. No element can both belong and not belong to A within U.
If |U|=20 and |A|=7, how large is A′?
Answer: 13. A and its complement partition U, so 20−7=13.
Discuss: Choose a universe of six symbols. Make a set and its complement, then check both partition rules.
Teaching note: A complement is not every imaginable object. It is relative to the stated universe. Open responses need discussion; they are not automatically marked.
Final check · Set 1
Try one set after the lessons. Use the other set after reviewing any errors.
How many elements are in {0}?
1. The set contains one element, the number 0.
Why is {2,9} not a subset of {2,4,6,8}?
9 is absent from the second set. One counterexample is enough to disprove inclusion.
Disjoint sets have…
No shared elements. Their intersection is empty.
If |A|=8, |B|=5 and their overlap has 3 members, the union has…
10. 8+5−3=10.
A∪A′ equals…
U. Every element of U is either in A or outside it.
If U grows while A stays fixed, can A′ change?
Yes. New universal-set elements outside A join its complement.
Final check · Set 2
Try one set after the lessons. Use the other set after reviewing any errors.
Which describes the same set as {1,3,5}?
{5,1,3,3}. Order and repeated listing do not change membership.
The empty set is a subset of…
Every set. It has no element that could violate the requirement for being a subset.
If A={a,b}, B={c,d}, their intersection is…
Empty. No member belongs to both.
Why subtract the intersection in the union count?
It was counted twice. Adding both set sizes counts the overlap twice; subtracting it once corrects that.
For A={1,2,3}, B={3,4}, A−B equals…
{1,2}. Keep elements in A that are not in B.
Taking a complement twice, within the same U, gives…
The original set. An element excluded by the first complement is restored by the second.
A chapter companion covering membership, subsets, union, intersection and complement with finite examples. Infinite sets, intervals and the full textbook exercise map still need further lessons.