At a glance. One lesson of about 60 minutes. It assumes listing a set and counting its elements (Unit 1). Students read and write A ∪ B, A ∩ B and A′, count elements of combined sets by listing, and use De Morgan's laws.
Calculator: no calculator. None is needed. Every set is small enough to list, and listing is the reasoning being taught.
What students will learn
Know
- The universal set U, and the symbols ∪, ∩ and A′.
- n(A) is the number of elements of A, and n(A′) = n(U) − n(A).
- De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.
Understand
- Union is inclusive: an element in both sets is in the union once.
- A complement only means something relative to U.
- A symbol statement and a shaded Venn region are two ways of saying the same thing.
Do
- Translate between words, symbols and shaded regions.
- Find A ∪ B, A ∩ B and A′ for sets given as lists or rules.
- Use a De Morgan law to rewrite a set before counting.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: correct use of ∪, ∩, ′ and n( ) in familiar sets (questions 1 to 8).
- Criterion B. Investigating patterns: the second figure. Ask students to predict whether two expressions will shade the same regions before pressing a button, and to say why the third pair fails.
- Criterion C. Communicating: moving between words, symbols and shading (question 4, question 9 and the warning card).
- Criterion D. Applying mathematics in real-life contexts: the hook's club lists, and what “or” means when a head teacher says it.
The one thing to do with the figure
Make students commit before they slide. In the first figure, ask “Which numbers are in A′ ∩ B?” and have pairs write the list before the slider moves. Then slide to A′ ∩ B and check.
In the second figure, press the third pair last. A pair that looks like a law but is not one makes the point that a Venn diagram is a way to test a statement, not just to illustrate one.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: collect predictions about the two club lists; do not settle them yet. |
| 12 min | First figure and key idea: each expression in turn, with the numbers in the list. Settle the hook. |
| 10 min | Second figure, the warning card and the worked example; pairs finish the faded example. |
| 5 min | The check. If many choose B, return to the overlap in the first figure. |
| 16 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | {2, 3, 4, 5, 6}; 1, 2, 3, 6, 9, 18; no (51 = 3 × 17). |
|---|---|
| Hook | A student in both appears once on the “or” list and once on the “and” list; the “and” list is never longer; they are equal only when everyone in either club is in both. |
| Worked example | S ∩ O = {1, 9}; S ∪ O = {1, 3, 4, 5, 7, 9, 11, 13, 15, 16, 17}; n(S′) = 14. |
| Faded example | S ∪ R = {1, 3, 4, 6, 9, 12, 15, 16, 18}, n = 9. |
| 1. What a union means | A. B reads ∪ as ∩; C describes M ∩ N′; D describes the complement of the union. |
| 2. Counting an intersection | 7. The primes up to 20 are 8 numbers; only 2 is not odd. |
| 3. Counting a complement | 12. 20 − 8; 1 is not prime, so it is in A′. |
| 4. From symbols to words | C. A swaps which set has the complement; B reads ∩ as ∪; D is the complement of the union. |
| 5. Intersection of multiples | 2. C ∩ D = {12, 24}. |
| 6. Complement of a union | 20. n(C ∪ D) = 10, so 30 − 10. |
| 7. De Morgan's law | B. A and D are the first law, not the second; C keeps the overlap. |
| 8. Using De Morgan's law to count | 16. n(E ∪ F) = 20 + 8 − 4 = 24; 40 − 24. |
| 9. From words to symbols | D. A swaps the sets; B is the complement of the overlap; C uses ∪ instead of ∩. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Reads ∪ as ∩, or describes a complement instead. | Ask: is a student who walks and cycles in M ∪ N? Then which symbol says “only”? |
| 8, 10 or 11 in question 2 | Counted one set, or counted the union. | Ask for the list of A ∩ B. Which of the primes is not in the list of odd numbers? |
| 8 or 11 in question 3 | Gave n(A), or treated 1 as prime. | Write U in a row and circle the primes. What is left over is A′. |
| A, B or D in question 4 | Swapped the complemented set, used the wrong connective, or took the complement of the union. | Test each option with the number 9: is it in A′, in B, in both? |
| 7, 5 or 1 in question 5 | Counted one set, or stopped at the first common multiple. | Write both lists in two rows and draw lines between the matches. |
| 10, 18 or 23 in question 6 | Gave the union, double counted the overlap, or complemented one set. | Make a two-circle Venn diagram and fill 12 and 24 first. |
| A, C or D in question 7 | Confuses the two laws, or keeps the overlap. | Shade (A ∩ B)′ on a two-circle diagram, then shade each option and compare. |
| 36, 12 or 24 in question 8 | Used the wrong law, double counted, or did not complement. | Rewrite E′ ∩ F′ as (E ∪ F)′ before counting anything. |
| A, B or C in question 9 | Complemented the wrong set, complemented the overlap, or used ∪. | Say it aloud: “in the choir” (C) and “not in the band” (B′). Two conditions at once means ∩. |
Questions worth asking
- Why must a set A′ be described relative to a universal set? Listen for “not in A” having no meaning until we say what else is allowed: the integers, the students in a school, and so on.
- When is A ∪ B the same as A ∩ B? Only when A = B. Then both are just A.
- Is n(A ∪ B) ever less than n(A)? Never: A ∪ B contains all of A, so it has at least n(A) elements.
- How could you test a statement like (A ∩ B)′ = A′ ∩ B′ without a proof? Shade both sides on a Venn diagram, or try small sets. One counterexample is enough to show it false.
Differentiation
A way in for anyone stuck
- Give a printed two-circle diagram with the four regions numbered 1 to 4, and have students write the region numbers for each expression.
- Colour A in one colour and B in another, so that the overlap has both colours and the outside has none.
- Always read each expression aloud in words before writing the elements.
Stretch
- Use the second figure to decide whether A ∪ (A ∩ B) always equals A. Explain with regions.
- Describe in words the set (A ∩ B′) ∪ (A′ ∩ B), and find a name for it: the elements in exactly one of the two sets.
Students learning in English as an additional language
“Union” means joining together, “intersection” is where two roads cross, and “complement” is what completes a set to the whole. Say the symbols aloud every time: ∪ “union” or “or”, ∩ “intersection” or “and”, ′ “prime” or “not”.
| Word | Plain meaning | Example |
|---|---|---|
| universal set | the set of everything being talked about; written U | all the students in a school year |
| union | the set of elements that are in A, in B or in both; written A ∪ B | the students in the choir or the band |
| intersection | the set of elements that are in both A and B; written A ∩ B | the students who do both |
| complement | the set of elements of U that are not in A; written A′ | everyone in the year who is not in the choir |
| Venn diagram | a picture of sets as overlapping circles inside a box for U | two circles with an overlap |
| De Morgan's laws | two rules that swap ∪ and ∩ when a complement is taken | (A ∪ B)′ = A′ ∩ B′ |
Sentence starters for explaining aloud or in writing:
- … is in A ∩ B because it is in … and in ….
- A′ is the set of everything in U that ….
- The two shadings are the same / different because ….
Two things not to say
“Union means add.” Adding counts an element in both sets twice; a union lists it once.
“The complement is the opposite set.” It is everything in U that is not in the set, and it changes if U changes.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- ListingU = {1, 2, …, 10}, A = {2, 4, 6, 8, 10}, B = {1, 2, 3, 4, 5}. List A ∩ B and A ∪ B.
A ∩ B = {2, 4}; A ∪ B = {1, 2, 3, 4, 5, 6, 8, 10}. - ComplementIn the same U, list B′.
B′ = {6, 7, 8, 9, 10}.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextIn a year group, J is the set of students who study Japanese and K the set who study Korean. Write in symbols: the students who study neither.
(J ∪ K)′, or J′ ∩ K′. - Earlier work (Unit 1)List {x | x ∈ ℕ, 2x < 9}.
{0, 1, 2, 3, 4}.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningProve with regions that A′ ∪ (A ∩ B) = A′ ∪ B.
Both shade everything outside A and, inside A, only the overlap: the region of B within A is A ∩ B. - Working backwardsn(U) = 25, n(A) = 15, n(B) = 12 and n((A ∪ B)′) = 3. Find n(A ∩ B).
n(A ∪ B) = 22, so n(A ∩ B) = 15 + 12 − 22 = 5.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to the slider or buttons, then the arrow keys or Enter) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Draw a two-circle diagram on the board and have students place number cards in the four regions.