Sets — Notes, Examples & Venn Diagrams

1. What is a set?

A set is a well-defined collection of distinct objects, called elements or members. Sets are written with capital letters and elements are listed in curly brackets, e.g. A = {1, 2, 3}.

2. Common notation

• n(A) — number of elements in A. Example: if A = {2,4,6,8}, n(A) = 4.
• x ∈ A — "x is an element of A". Example: 3 ∈ {1,3,5}.
• x ∉ A — "x is not an element of A".
• A' — complement of A in the universal set (elements not in A).
• ∅ — the empty set (no elements).
• 𝓔 — universal set (all items under consideration).
• A ⊆ B — A is a subset of B.
• A ∪ B — union: elements in A or B or both.
• A ∩ B — intersection: elements common to A and B.

3. Example set definitions

• A = {x : x is a natural number} — natural numbers {1,2,3,...}.
• B = {(x,y) : y = mx + c} — points on the straight line y = mx + c.
• C = {x : a ≤ x ≤ b} — all numbers between a and b inclusive.
• D = {a, b, c, ...} — explicitly listed elements.

4. Venn diagram — worked 2-set example

Problem: In a class of 50 students, 33 like English, 42 like Math and 3 like neither. Use a Venn diagram and find the breakdown.

Solution (using formula):
Students liking at least one subject = 50 − 3 = 47.
Use: n(E ∪ M) = n(E) + n(M) − n(E ∩ M).
So 47 = 33 + 42 − n(E ∩ M) ⇒ n(E ∩ M) = 28.

English only = 33 − 28 = 5. Math only = 42 − 28 = 14. Both = 28. Neither = 3.

Probability using the diagram

1. P(likes English) = 33/50 = 0.66
2. P(likes Math) = 42/50 = 0.84
3. P(likes both) = 28/50 = 0.56
4. P(likes at least one) = 47/50 = 0.94
5. P(likes neither) = 3/50 = 0.06

5. Additional probability examples (practice)

Example A: If a student is chosen at random from the class, what is the probability they like exactly one of the two subjects?

Solution: Exactly one = English only + Math only = 5 + 14 = 19 ⇒ P = 19/50 = 0.38.

Example B: What is P(not English)?

Solution: Not English = total − English = 50 − 33 = 17 ⇒ P = 17/50 = 0.34.

6. 3-set Venn diagram (example + SVG)

Use a 3-set diagram when three properties are considered, e.g., English (E), Math (M) and Science (S). Below is a small example to practice intersection/union reasoning.

Example: In a group of 80 students:
n(E) = 40, n(M) = 50, n(S) = 35,
n(E ∩ M) = 20, n(E ∩ S) = 10, n(M ∩ S) = 15,
n(E ∩ M ∩ S) = 5, and the rest like none of these subjects.

Find students who like at least one subject.

Solution: Use inclusion-exclusion:
n(E ∪ M ∪ S) = n(E) + n(M) + n(S) − n(E∩M) − n(E∩S) − n(M∩S) + n(E∩M∩S)
= 40 + 50 + 35 − 20 − 10 − 15 + 5 = 85 − 45 + 5 = 45.
So 45 students like at least one of the three. Therefore 80 − 45 = 35 like none.

7. Tips for students

• Always identify the universal set before you begin.
• Label every region (e.g., "English only", "both", "neither").
• Use the inclusion–exclusion formula for unions when more than two sets are involved.
• Check totals: sum of all disjoint regions + neither should equal the universal set size.

8. Quick formulas (memorise)

• n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
• n(A ∪ B ∪ C) = n(A)+n(B)+n(C) − n(A∩B) − n(A∩C) − n(B∩C) + n(A∩B∩C)
• P(A) = n(A) / n(Universal)

Abel Masitsa.