Sets and Venn Diagrams
What is a Set?
A set is a well-defined collection of distinct objects, called elements or members. Sets are written with curly braces, and each element is listed once.
Examples:
- A = {2, 4, 6, 8} โ even single-digit numbers
- B = {cat, dog, rabbit} โ some common pets
- C = {} โ the empty set (written โ ), which contains no elements
The order of elements inside a set does not matter: {1, 2, 3} is the same set as {3, 1, 2}.
Set Notation
| Symbol | Meaning | Example |
|---|---|---|
| โ | is an element of | 4 โ {2, 4, 6} |
| โ | is not an element of | 5 โ {2, 4, 6} |
| n(A) | number of elements in A | n({2, 4, 6}) = 3 |
| โ or {} | empty set | no elements |
| ฮพ | universal set | all elements under consideration |
| โ | is a subset of | {2, 4} โ {2, 4, 6} |
A is a subset of B (written A โ B) if every element of A is also in B. Every set is a subset of itself, and the empty set is a subset of every set.
Union and Intersection
Union (A โช B) โ everything in A or B (or both). Think of it as combining the two sets.
Intersection (A โฉ B) โ everything in both A and B. Think of it as the overlap.
Example: A = {1, 2, 3, 4} and B = {3, 4, 5, 6}
- A โช B = {1, 2, 3, 4, 5, 6} โ all elements from either set, listed once
- A โฉ B = {3, 4} โ only the elements appearing in both
When two sets share no elements, they are called disjoint and A โฉ B = โ .
Complement
The complement of A, written Aโฒ, contains every element in the universal set ฮพ that is not in A.
Example: ฮพ = {1, 2, 3, 4, 5, 6, 7, 8}, A = {2, 4, 6, 8} Aโฒ = {1, 3, 5, 7}
Notice that A and Aโฒ together make up the entire universal set, and they share no elements: A โฉ Aโฒ = โ .
Venn Diagrams
A Venn diagram uses overlapping circles inside a rectangle to show the relationships between sets. The rectangle represents the universal set ฮพ.
ฮพ
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
โ โโโโโโโโโโโโ โโโโโโโโโโโโ โ
โ โ A only โโฉ โ B only โ โ
โ โ {1, 2} โ โ {5, 6} โ โ
โ โ {3, 4} โ โ
โ โโโโโโโโโโโโ โโโโโโโโโโโโ โ
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
- Left region (A only): elements in A but not B โ {1, 2}
- Overlapping region: elements in both A and B โ A โฉ B = {3, 4}
- Right region (B only): elements in B but not A โ {5, 6}
- Outside both circles: elements in ฮพ but in neither A nor B
Venn diagrams with three sets work the same way โ three overlapping circles create seven distinct regions (each subset combination).
The Counting Formula
When two sets overlap, simply adding their sizes double-counts the shared elements. The correct formula is:
n(A โช B) = n(A) + n(B) โ n(A โฉ B)
Example: In a class of 30 students, 18 study French, 15 study Spanish, and 8 study both.
n(French โช Spanish) = 18 + 15 โ 8 = 25 students study at least one language.
That means 30 โ 25 = 5 students study neither.
This formula appears frequently in exam problems where you are given totals and need to find an unknown region.
Solving Venn Diagram Problems
A common exam style: you are given totals for each region and asked to find a missing value.
Example: ฮพ has 40 elements. n(A) = 22, n(B) = 18, n(A โฉ B) = 10.
Step 1 โ Fill in the intersection: A โฉ B = 10. Step 2 โ A only: 22 โ 10 = 12. B only: 18 โ 10 = 8. Step 3 โ Elements in A โช B: 12 + 10 + 8 = 30. Step 4 โ Neither: 40 โ 30 = 10.
Draw the diagram, fill each region in order, and use the total to find anything remaining.
Useful Facts to Remember
- n(A) + n(Aโฒ) = n(ฮพ) โ a set and its complement together fill the universal set
- A โฉ A = A and A โช A = A โ a set intersected or united with itself is just itself
- A โฉ โ = โ and A โช โ = A โ the empty set contributes nothing to a union and absorbs the intersection
- (Aโฒ)โฒ = A โ the complement of a complement returns the original set
Common Mistakes
- Listing elements twice in a union: each element appears once, no matter how many sets it belongs to.
- Confusing โช and โฉ: โช (union) looks like a cup that holds all the water; โฉ (intersection) looks like a cap covering only the middle part.
- Forgetting to subtract the overlap: when using n(A โช B) = n(A) + n(B) โ n(A โฉ B), the subtraction is essential whenever A and B share elements.
- Placing elements in the wrong Venn region: elements in A โฉ B go in the overlapping middle section, not in one circle alone.
Questions & comments
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