Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 10, Unit 1: list {x | x ∈ ℤ, 2 ≤ x < 7}.
Answer
{2, 3, 4, 5, 6}. 2 is in because of ≤; 7 is not, because of <. - From Year 8: list the factors of 18.
Answer
1, 2, 3, 6, 9 and 18. - From Year 9: is 51 a prime number?
Answer
No. 51 = 3 × 17, so it has a factor other than 1 and itself.
Words used in this lesson
- universal set
- the set of everything being talked about; written U. For example: all the students in a school year.
- union
- the set of elements that are in A, in B or in both; written A ∪ B. For example: the students in the choir or the band.
- intersection
- the set of elements that are in both A and B; written A ∩ B. For example: the students who do both.
- complement
- the set of elements of U that are not in A; written A′. For example: everyone in the year who is not in the choir.
- Venn diagram
- a picture of sets as overlapping circles inside a box for U. For example: two circles with an overlap.
- De Morgan's laws
- two rules that swap ∪ and ∩ when a complement is taken. For example: (A ∪ B)′ = A′ ∩ B′.
Clubs, choirs and who is counted twice
At a school in Accra some students sing in the choir, some play in the band, and a few do both. The head teacher asks for a list of everyone who is “in the choir or in the band”, and a second list of everyone who is “in the choir and in the band”.
Predict first: does a student who does both appear once, twice or not at all on the first list? Which list can never be longer than the other? Could the two lists ever be the same length? Then use the first figure to test your ideas on twelve numbers.
A ∪ B = {2, 3, 4, 6, 8, 9, 10, 12}: in A, in B or in both. Elements in both are listed once. n(A ∪ B) = 8.
Show me the steps
- A ∪ B = {2, 3, 4, 6, 8, 9, 10, 12}: every number in A, in B or in both. 6 and 12 are in both, and each is listed once. n(A ∪ B) = 8.
- Slide to A ∩ B: only the numbers in both circles. A ∩ B = {6, 12}, so n(A ∩ B) = 2.
- Slide to A′: every number in U that is not in A. A′ = {1, 3, 5, 7, 9, 11}, so the 3 and 9 from B are in A′ because they are odd.
- Slide to (A ∪ B)′: the numbers outside both circles. (A ∪ B)′ = {1, 5, 7, 11}, so n((A ∪ B)′) = 4.
Three symbols name the regions of a Venn diagram.
- U, the universal set: everything under discussion. Every other set sits inside it.
- A ∪ B, the union: in A, or in B, or in both. Read “A union B” or “A or B”.
- A ∩ B, the intersection: in both A and B. Read “A intersection B” or “A and B”.
- A′, the complement: every element of U that is not in A. Read “A prime” or “not A”.
De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′. Take the complement of a union and the union turns into an intersection of complements, and the other way round.
(A ∪ B)′ and A′ ∩ B′ shade exactly the same regions, so they are the same set.
Show me the steps
- (A ∪ B)′ and A′ ∩ B′: both shade only the region outside both circles. They are the same set, which is De Morgan's first law.
- Press (A ∩ B)′ and A′ ∪ B′: both shade three regions, everything except the overlap. They are the same set, De Morgan's second law.
- Press A′ ∪ B and A′ ∩ B: three regions on the left and only one on the right. They are not the same set, so no law makes them equal.
The complement depends on U. A′ means “in U but not in A”, so the same set A has a different A′ if the universal set changes. And the bracket matters: (A ∪ B)′ means “in neither”, but A′ ∪ B′ means “not in A, or not in B”, which is a much bigger set.
Worked example: reading set symbols
U = {x | x ∈ ℤ, 1 ≤ x ≤ 18}, S = {1, 4, 9, 16} (the squares) and O = the odd numbers in U. Find S ∩ O, S ∪ O and n(S′).
- List O: {1, 3, 5, 7, 9, 11, 13, 15, 17}.
- Numbers in both lists: 1 and 9. So S ∩ O = {1, 9}.
- Everything in either list, each number once: S ∪ O = {1, 3, 4, 5, 7, 9, 11, 13, 15, 16, 17}.
- S has 4 elements and U has 18, so S′ has 18 − 4 = n(S′) = 14.
Your go. The first step is given. Using the same U and S, let R = {3, 6, 9, 12, 15, 18} (the multiples of 3). List S ∪ R and find n(S ∪ R).
- Write both lists: S = {1, 4, 9, 16} and R = {3, 6, 9, 12, 15, 18}. 9 is in both lists.
- Now write every number from either list once, in order, and count them.
Check the answer
S ∪ R = {1, 3, 4, 6, 9, 12, 15, 16, 18}, so n(S ∪ R) = 9. A common slip is to count 4 + 6 = 10 and list 9 twice.
Check before you go on
1. M is the set of students who walk to school and N is the set of students who cycle. Which statement describes M ∪ N?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. U = {x | x ∈ ℤ, 1 ≤ x ≤ 20}. A is the set of prime numbers in U and B is the set of odd numbers in U. How many elements does A ∩ B have?
3. With the same U and A (the primes up to 20), how many elements does A′ have?
Level 2 · Building
4. Using U, A (primes) and B (odd numbers) from Questions 2 and 3, which description matches A′ ∩ B?
5. U = {x | x ∈ ℤ, 1 ≤ x ≤ 30}, C = the multiples of 4 and D = the multiples of 6. What is n(C ∩ D)?
6. With the same U, C and D, what is n((C ∪ D)′)?
Level 3 · Stretch
7. Which set is always equal to (A ∩ B)′?
8. U = {x | x ∈ ℤ, 1 ≤ x ≤ 40}, E = the multiples of 2 and F = the multiples of 5. How many elements does E′ ∩ F′ have?
Something to think about
A student claims that (A ∪ B)′ is the same as A′ ∪ B′ because the symbols look almost identical. Find a set A and a set B inside a small universal set that proves the claim wrong, then say which of the two De Morgan laws it breaks.
One way to think about it
Try U = {1, 2, 3, 4}, A = {1, 2} and B = {2, 3}. Then (A ∪ B)′ = {4}, but A′ = {3, 4} and B′ = {1, 4}, so A′ ∪ B′ = {1, 3, 4}. The two sets differ, because A′ ∪ B′ equals (A ∩ B)′, not (A ∪ B)′. The complement swaps the sign: ∪ becomes ∩ and ∩ becomes ∪.
Exit ticket
9. C is the set of students in the choir and B is the set of students in the band. Which symbols describe the students who are in the choir but not in the band?