Intersection — A ∩ B ∩ C
All 7 regions of a 3-set Venn diagram labelled
A only — A − (B ∪ C)
Elements in A but in neither B nor C
Complement — Ā
Everything in 𝕌 outside A (B, C may exist)
Cardinality of set A, written n(A) or
|A|, is the number of distinct elements in
A.
Example: A = {a,b,c,d} → n(A) = 4
Example: A = {a,b,c,d} → n(A) = 4
Key Rules — Three Sets
1
Count each element only once. {1,1,2,3} has cardinality 3.
2
2-set union: n(A∪B) = n(A)
+ n(B) − n(A∩B)
3
3-set inclusion-exclusion:
n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C)
n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C)
4
Complement: n(Ā) = n(𝕌) −
n(A)
5
Disjoint sets (no overlap): n(A∩B) = 0, so n(A∪B) =
n(A)+n(B)
6
A-only region: n(A only) =
n(A) − n(A∩B) − n(A∩C) + n(A∩B∩C)
7
Elements in none: n(𝕌) −
n(A∪B∪C)
Basic Cardinality of a Single Set
A = {3, 7, 11, 15, 19}
Find n(A).
👀Elements: 3, 7, 11, 15, 19
🔢Here the set contains
5 elements, so n(A) = 5
Answer
n(A) = 5
n(A∪B) using 2-set formula
A={1,2,3,4,5}B={4,5,6,7}
📊n(A)=5, n(B)=4
🔍A∩B = {4,5} → n(A∩B)=2
🧮5+4−2 = 7
Answer
n(A∪B) = 7
3-Set Inclusion-Exclusion
n(A)=20n(B)=18n(C)=15
n(A∩B)=5n(B∩C)=4n(A∩C)=6n(A∩B∩C)=2
➕Sum: 20+18+15 = 53
➖Subtract pairwise:
53−(5+4+6) = 38
➕Add triple: 38+2 = 40
Answer
n(A∪B∪C) = 40
Question 1
A={1,2,3,4}, B={3,4,5,6}, C={4,6,7,8}. Find
n(A∪B∪C).
Hint:
n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(A∩C)+n(A∩B∩C)
Step 1: What is the formula for n(A∪B∪C)?
✓ Final Answer
n(A∪B∪C) = 8
Hint: Find elements common to all three sets.
A∩B = elements in both A and B?
✓ Final Answer
n(A∩B∩C) = 1 — only {4}
Hint: Apply inclusion-exclusion: sum − pairwise +
triple.
Step 1: n(A)+n(B)+n(C) = ?
✓ Final Answer
n(A∪B∪C) = 18
Hint: A only = A − (B∪C). Remove anything A shares
with B or C.
Step 1: Count B ∪ C from the sets or Venn = ?
✓ Final Answer
n(A only) = 2 — elements {1, 2}
Hint: n(Ā) = n(𝕌) − n(A). List multiples of 3 up to
15 first.
A = {3,6,9,12,15}. What is n(A)?
✓ Final Answer
n(Ā) = 10
Hint: Apply 3-set inclusion-exclusion step by step.
Step 1: n(A)+n(B)+n(C) = ?
✓ Final Answer
n(A∪B∪C) = 47 · Neither = 3
Hint: Rearrange: n(A∩B∩C) = n(A∪B∪C) − n(A) − n(B) −
n(C) + n(A∩B) + n(B∩C) + n(A∩C)
Step 1: n(A)+n(B)+n(C) = ?
✓ Final Answer
n(A∩B∩C) = 3
Hint: Find n(A∪B∪C) first, then None = n(𝕌) −
n(A∪B∪C).
Step 1: n(A)+n(B)+n(C) = ?
✓ Final Answer
n(A∪B∪C)=47 · None = 13