De Morgan's Laws

Compact study note.

Summary

De Morgan's laws describe how complements distribute over unions and intersections. They are used constantly when translating probability statements involving 'not', 'and', and 'or'.^[1]

Prerequisites

• Sample Space and Events

Notation and Assumptions

Complements are taken relative to $\mathrm{\Omega }$ :

Essential Result

De Morgan identities:

The same identities hold for countable families.

Small Example

If events are 'rain' and 'wind', then 'neither rain nor wind' is

Common Mistakes

• Confusing 'not both' with 'neither'.
• Forgetting that complements depend on the chosen sample space.

Connections

• Sigma-Algebra
• Conditional Probability
• Independence

References