Set Operations: Union, Intersection, Complement

Problem 1.2 • Understanding set theory in probability with De Morgan's Laws

∪

Union (A ∪ B)

Elements in A OR B (or both)

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

The union includes all outcomes that belong to at least one of the sets.

∩

Intersection (A ∩ B)

Elements in BOTH A AND B

P(A ∩ B) ≤ min(P(A), P(B))

The intersection can never exceed the probability of either individual set.

−

Difference (A − B)

Elements in A but NOT in B

P(A − B) = P(A) − P(A ∩ B)

Remove the overlapping region from set A.

ᶜ

Complement (Aᶜ)

Everything NOT in A

P(Aᶜ) = 1 − P(A)

The complement contains all outcomes outside the set.

⚡ De Morgan's Laws

These laws show how complements distribute over unions and intersections:

(A ∪ B)ᶜ
=
Aᶜ ∩ Bᶜ

"NOT (A or B)" is the same as "(NOT A) and (NOT B)"

(A ∩ B)ᶜ
=
Aᶜ ∪ Bᶜ

"NOT (A and B)" is the same as "(NOT A) or (NOT B)"

🎮 Interactive Set Explorer

Click cells to toggle membership. Watch how different operations change the selection.

Selected elements: 0 / 100
P(result) = 0.00