Learning Objectives
- Use set notation accurately, including union, intersection, complement, subset and number of elements.
- Represent and interpret relationships involving two or three sets with Venn diagrams.
- Use the inclusion–exclusion principle to calculate missing numbers.
- Solve word problems by placing information in the correct Venn regions.
- Distinguish between elements, subsets, empty regions and the universal set.
Key Terms
- Universal set
- The complete set of objects being considered, usually denoted by \xi or described in words.
- Intersection
- A\cap B, the elements that belong to both A and B.
- Union
- A\cup B, the elements that belong to A or B or both.
- Complement
- A\prime, the elements in the universal set that are not in A.
- Subset
- A\subseteq B means every element of A is also an element of B.
- Cardinality
- The number of elements in a set, written n(A).
- Disjoint sets
- Sets with no common elements, so A\cap B=\varnothing.
What This Chapter Covers
- Set language and notation required by Cambridge 4024
- Two-set and three-set Venn diagrams
- Inclusion–exclusion and missing-region calculations
- Word problems involving categories and survey data
- Logical interpretation of complements and combined regions
Reading Set Notation Precisely
A set is a well-defined collection of distinct objects. The objects are called elements. Cambridge questions may describe a set by listing its elements, by using a rule, or by showing it as a region in a Venn diagram. Curly brackets are used for listed elements, for example A=\{2,4,6,8\}. Repetition does not create extra elements: \{1,1,2,2,3\}=\{1,2,3\}.
The symbol \in links an element to a set, while \subseteq links one set to another. Therefore 3\in A and \{2,4\}\subseteq A are meaningful, but 3\subseteq A is normally not.
| Notation | Meaning | Example |
|---|---|---|
| x\in A | x is an element of A | 4\in\{2,4,6\} |
| x\notin A | x is not an element of A | 5\notin\{2,4,6\} |
| A\subseteq B | Every element of A lies in B | \{2,4\}\subseteq\{2,4,6\} |
| A\cup B | In A or B or both | Union contains every element appearing in either set |
| A\cap B | In both A and B | Intersection is the overlap |
| A\prime | Not in A but inside the universal set | The region outside A |
| n(A) | Number of elements in A | If A has five distinct elements, n(A)=5 |
Two-Set Venn Diagrams
When information is given about two sets, begin with the intersection because those objects belong to both categories. Next fill the parts belonging only to A and only to B. Finally place the number belonging to neither outside both circles but inside the universal rectangle.
Reliable Order For A Two-Set Problem
- Write the total number in the universal set.
- Place the value for A\cap B in the overlap.
- Calculate A only by subtracting the overlap from n(A).
- Calculate B only by subtracting the overlap from n(B).
- Add all values inside the circles and subtract from the total to find neither.
- Check that every region is non-negative and the regions sum to the universal total.
The intersection is subtracted once because it was counted in both n(A) and n(B).
Worked Example: Survey With Two Sets
Question: In a class of 40 students, 23 study French, 18 study German and 9 study both. Find the number who study only French, only German, at least one language and neither language.
- French only =23-9=14.
- German only =18-9=9.
- At least one =14+9+9=32, or 23+18-9=32.
- Neither =40-32=8.
Answer: French only 14, German only 9, at least one 32, neither 8.
Three-Set Venn Diagrams
A three-set diagram has eight regions inside the universal rectangle: the central triple intersection, three pairwise-only intersections, three single-set-only regions and the region outside all sets. The phrase “A and B” may include people also in C unless the question says “A and B only”. This distinction is essential.
Order For Three Sets
- Fill the central region A\cap B\cap C first.
- Fill each pairwise-only region by subtracting the central region from the given pairwise total.
- Fill each single-only region by subtracting every already placed part of that set from its total.
- Add all seven regions inside the circles.
- Subtract from the universal total to find the number in none of the sets.
Worked Example: Three Activities
Question: Of 60 students, 30 play cricket, 28 play football and 20 play hockey. Twelve play cricket and football, 8 play cricket and hockey, 7 play football and hockey, and 5 play all three. Find the number who play only cricket and the number who play at least one sport.
- Cricket and football only =12-5=7.
- Cricket and hockey only =8-5=3.
- Football and hockey only =7-5=2.
- Cricket only =30-(7+3+5)=15.
- Use inclusion–exclusion: 30+28+20-12-8-7+5=56.
Answer: Only cricket 15; at least one sport 56.
Complements And De Morgan-Style Reading
The complement sign applies to the whole expression immediately before it. (A\cup B)\prime means neither A nor B. (A\cap B)\prime means not in both simultaneously; it includes A only, B only and neither. Even where a formal law is not requested, the ability to shade or describe these regions is important.
| Expression | Region described |
|---|---|
| A\prime\cap B | In B but not in A |
| A\cap B\prime | In A but not in B |
| (A\cup B)\prime | In neither A nor B |
| (A\cap B)\prime | Everywhere except the overlap of A and B |
| A\cup B\prime | In A, or outside B, or both |
Using Algebra In Venn Diagrams
Regions may contain expressions such as x+3 or 2x-1. Add all relevant regions and form an equation from the stated total. After solving, substitute back into every region and check that no frequency is negative.
Worked Example: Algebraic Regions
Question: A universal set has 50 elements. A two-set diagram contains x+4 in A only, 2x in the intersection, x-1 in B only and 7 in neither. Find x.
- Add all regions: (x+4)+2x+(x-1)+7=50.
- Simplify: 4x+10=50.
- Solve: 4x=40, so x=10.
- Check the regions: 14, 20, 9 and 7 sum to 50.
Answer: x=10.
Examination Guidance
- Always distinguish “both” from “both only” in a three-set problem.
- Put intersections in first; otherwise the same people may be counted twice.
- Use n(A\cup B) for “A or B or both”.
- Check that every region total is a whole number and that all regions sum to the universal total.
- When shading, follow brackets and complement signs before deciding which region is required.
Common Mistakes
- Adding n(A)+n(B) without subtracting the intersection.
- Putting the full pairwise intersection into the pairwise-only region when a triple intersection exists.
- Confusing an element with a subset.
- Leaving the “neither” region unaccounted for.
- Treating “or” as exclusive; in set notation union normally includes the overlap.
Knowledge Check And Practice
1. If n(A)=18, n(B)=25 and n(A\cap B)=7, find n(A\cup B).
2. In a group of 50, 32 like tea, 21 like coffee and 15 like both. How many like neither?
3. Describe A\prime\cap B in words.
4. If A\subseteq B, what is A\cap B?
5. If two sets are disjoint, what is their intersection?
6. A three-set diagram has 6 in the central region and a pairwise total of 14 for A and B. What goes in the A-and-B-only region?
