1 Basic Set-Theoretic Notations (Events)
| Term / Statement | Set Notation | Meaning (in probability language) |
|---|---|---|
| Sample space | S |
Set of all possible outcomes. |
| Outcome (sample point) | ω (or a) |
A single outcome of the random experiment. |
| Event A | A ⊂ S (or A ⊆ S) |
A is a subset of S. |
| Event A has occurred | ω ∈ A |
The observed outcome belongs to A. |
| Event A has not occurred | ω ∉ A |
The observed outcome does not belong to A. |
| At least one of A or B occurs | A ∪ B |
A occurs OR B occurs OR both occur. |
| Both A and B occur | A ∩ B |
Common outcomes of A and B. |
| Not A (complement of A) | Aᶜ or A′ or Ā |
All outcomes in S that are not in A. |
2 Union of Events
Union of A and B is written as A ∪ B.
A ∪ B = outcomes where A occurs, or B occurs, or both occur.
Example (Die)
Let S = {1,2,3,4,5,6}, A = {2,4,6} (even), B = {1,3,5} (odd).
A ∪ B = {1,2,3,4,5,6} = S
Example (Sets)
If A = {1,2,3}, B = {3,4}, then
A ∪ B = {1,2,3,4}
3 Intersection of Events
Intersection of A and B is written as A ∩ B.
A ∩ B = outcomes where both A and B occur (common outcomes).
Example (Die)
Let A = {2,4,6} (even) and C = {2,3,5}.
A ∩ C = {2}
Key idea
If there is no common outcome, then intersection is empty: ∅.
4 Complement of an Event
The complement of A is written as Aᶜ (or A′ or Ā).
Aᶜ = S \\ A
Example (Die)
S = {1,2,3,4,5,6}, A = {2,4,6}
Aᶜ = {1,3,5}
Important
A ∪ Aᶜ = S and A ∩ Aᶜ = ∅
5 De Morgan’s Laws
These connect complement with union / intersection:
(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ(A ∩ B)ᶜ = Aᶜ ∪ Bᶜ
Quick verification (Die)
A={2,4,6}, B={1,3,5}, S={1,2,3,4,5,6}
A ∪ B = S ⇒ (A ∪ B)ᶜ = ∅
Aᶜ = {1,3,5}, Bᶜ = {2,4,6}
Aᶜ ∩ Bᶜ = ∅ ✅
Interpretation
(A ∪ B)ᶜ means “neither A nor B occurs”.
Aᶜ ∩ Bᶜ means “not A AND not B”.
6 “A implies B” using subsets
If event A implies event B, then A is a subset of B:
A ⊆ B ⇒ whenever A occurs, B must also occur.
Example (Die)
Let A = “number is 4” = {4}, and B = “number is even” = {2,4,6}.
Then A ⊆ B, so occurrence of A guarantees occurrence of B.
7 Mutually Exclusive and Not Mutually Exclusive
Mutually exclusive events cannot occur together:
A ∩ B = ∅
Not mutually exclusive events can occur together (partially overlapping):
A ∩ B ≠ ∅
Example (Mutually exclusive)
Single card draw: A = “Heart”, B = “Spade”.
A ∩ B = ∅
Example (Not mutually exclusive)
Single card draw: A = “King”, B = “Spade”.
A ∩ B contains “King of Spades” ⇒ not empty.
8 Mutually Exclusive and Exhaustive System of Events
Events E₁, E₂, …, Eₙ form a mutually exclusive and exhaustive system if:
Eᵢ ∩ Eⱼ = ∅fori ≠ j(no overlap)E₁ ∪ E₂ ∪ … ∪ Eₙ = S(covers entire sample space)
Example (Die)
E₁={1}, E₂={2}, …, E₆={6}
All pairwise intersections are empty, and union is S.
Example (Three events)
Let S be all outcomes. If A ∩ B = ∅, B ∩ C = ∅, A ∩ C = ∅, and A ∪ B ∪ C = S, then {A,B,C} is mutually exclusive & exhaustive.
9 Worked Example (From your manuscript-style sets)
Take
S = {1,2,3,4,5,6}
A = {2,4,6}, B = {1,3,5}, C = {2,3,5}
A ∪ B = {1,2,3,4,5,6} = S
A ∩ C = {2}
B ∩ C = {3,5}
Aᶜ = {1,3,5}
Bᶜ = {2,4,6}
(A ∪ B)ᶜ = ∅ and Aᶜ ∩ Bᶜ = ∅ (De Morgan check)