Sets · Learner pages

Lesson 1 · Describe membership precisely

Represent a finite set and distinguish membership from inclusion.

Name or initials: ____________   Date: ____________

A mathematical set needs a clear rule deciding whether each candidate belongs.

1. Use a definite rule. Let A contain the positive even integers below 9. The rule gives A={2,4,6,8}.
2. Ignore order and repetition. {8,2,4,6,2} describes the same set: repeated writing does not create another element.
3. Distinguish two statements. 2∈A means that the number 2 belongs to A. {2}⊆A means that every member of the one-element set {2} belongs to A.
4. Test inclusion. B={2,8} is a subset of A because both its elements occur in A. C={2,9} is not, because 9 is a counterexample.

Worked example: If D={letters in MATH}, then D={M,A,T,H} and |D|=4. The sets {A,M} and the empty set are subsets of D; {A,S} is not because S∉D.

1. How many elements are in {2,2,5,7}?

   1. 4 · 2. 3 · 3. 2

2. If A={2,4,6}, which is true?

   1. 4∈A · 2. 5∈A · 3. {5}⊆A

3. Which is a subset of {a,b,c}?

   1. {a,d} · 2. {a,c} · 3. {d}

Discuss: Find a vague collection, rewrite its rule to remove opinion, and list its members.

Lesson 2 · Combine sets without double counting

Calculate union, intersection and the size of a union.

Name or initials: ____________   Date: ____________

Union uses either set; intersection requires membership in both.

1. Place each member once. Take A={1,2,3} and B={3,4,5}. The shared member 3 sits in the overlap of the diagram.
2. Collect the union. A∪B={1,2,3,4,5}. It includes both circles but lists the shared member only once.
3. Collect the intersection. A∩B={3}. Only membership in both sets qualifies; 1 and 4 do not.
4. Check the count. For finite sets, |A∪B|=|A|+|B|−|A∩B|. Here 3+3−1=5. Subtract the overlap once because adding both sizes counted it twice.

Worked example: In a class, 12 pupils join chess, 9 join art and 4 join both. The number joining at least one is 12+9−4=17. This alone does not tell us how many join neither; the class size is also needed.

1. If A={1,2}, B={2,3}, what is A∪B?

   1. {2} · 2. {1,2,3} · 3. {1,3}

2. For the same sets, A∩B is…

   1. {2} · 2. {1,3} · 3. {1,2,3}

3. If |A|=7, |B|=6 and |A∩B|=2, find |A∪B|.

   1. 15 · 2. 13 · 3. 11

Discuss: Create two sets with four members each whose union has six members. Explain the overlap.

Lesson 3 · Use the universe when taking a complement

Explain why a complement depends on the chosen universal set.

Name or initials: ____________   Date: ____________

A universal set specifies the objects currently under consideration.

1. State the universe. Let U={1,2,3,4,5,6} and A={1,2,3}. Without U, a complement is not fully specified.
2. Remove the members of A. The elements of U not in A are 4,5,6. Thus A′={4,5,6}.
3. Check the partition. A and A′ do not overlap, and their union is U. Their sizes add to |U|.
4. Change the universe carefully. If U is enlarged to {1,2,3,4,5,6,7}, A stays the same but its complement becomes {4,5,6,7}.

Worked example: Let U={a,b,c,d,e}, A={a,c}. Then A′={b,d,e}. The complement of A′ is A again. If B={c,d}, A−B={a}; subtraction of sets removes membership, not numerical values.

1. If U={1,2,3,4} and A={2,4}, A′ equals…

   1. {2,4} · 2. {1,3} · 3. {1,2,3,4}

2. A∩A′ is always…

   1. U · 2. A · 3. Empty

3. If |U|=20 and |A|=7, how large is A′?

   1. 27 · 2. 13 · 3. 7

Discuss: Choose a universe of six symbols. Make a set and its complement, then check both partition rules.

Final check · Set 1

Try one set after the lessons. Use the other set after reviewing any errors.

1. How many elements are in {0}?

   0 / 1 / Infinitely many

2. Why is {2,9} not a subset of {2,4,6,8}?

   It has only two members / Its members are numbers / 9 is absent from the second set

3. Disjoint sets have…

   No shared elements / Equal sizes / Every element shared

4. If |A|=8, |B|=5 and their overlap has 3 members, the union has…

   16 / 10 / 13

5. A∪A′ equals…

   U / Empty / A only

6. If U grows while A stays fixed, can A′ change?

   Yes / Never / Only if A is empty

Final check · Set 2

Try one set after the lessons. Use the other set after reviewing any errors.

1. Which describes the same set as {1,3,5}?

   {5,1,3,3} / {1,3} / {1,3,5,7}

2. The empty set is a subset of…

   No sets / Every set / Only itself

3. If A={a,b}, B={c,d}, their intersection is…

   {a,b,c,d} / {a,c} / Empty

4. Why subtract the intersection in the union count?

   It was counted twice / It never belongs to the union / It has negative size

5. For A={1,2,3}, B={3,4}, A−B equals…

   {1,2} / {4} / {1,2,3,4}

6. Taking a complement twice, within the same U, gives…

   U always / The original set / The empty set