A disjunction P \vee Q is a compound statement that is true whenever at least one of its parts is true. Disjunction expresses alternatives and is the logical backbone of union, piecewise definitions, and compound probability.
Definition
A disjunction P \vee Q is a compound statement that is true whenever at least one of its parts is true.
💡 Intuition
At least one must be true. Logical OR is inclusive — "P or Q or both" — unlike the exclusive everyday "either/or."
🎯 Core Idea
P \vee Q is false only when both P and Q are false; it is true in all three other cases (TT, TF, FT).
Example
Formula
Notation
P \vee Q
🌟 Why It Matters
Disjunction expresses alternatives and is the logical backbone of union, piecewise definitions, and compound probability.
💭 Hint When Stuck
Compare with AND by asking: 'Does at least one part need to be true, or do both?' If at least one, you want OR.
Formal View
Related Concepts
🚧 Common Stuck Point
Logical OR is inclusive (includes 'both'). 'XOR' is exclusive (one but not both).
⚠️ Common Mistakes
- Using exclusive-or reasoning — in logic, P \vee Q is true when both are true, unlike everyday 'or'
- Thinking P \vee Q means exactly one is true — that is XOR (P \oplus Q), not OR
- Confusing \vee (or) with \wedge (and) — P \vee Q is false ONLY when both are false
Go Deeper
Frequently Asked Questions
What is Disjunction in Math?
A disjunction P \vee Q is a compound statement that is true whenever at least one of its parts is true.
What is the Disjunction formula?
P \vee Q is false \Leftrightarrow P is false and Q is false
When do you use Disjunction?
Compare with AND by asking: 'Does at least one part need to be true, or do both?' If at least one, you want OR.
Prerequisites
Next Steps
Cross-Subject Connections
How Disjunction Connects to Other Ideas
To understand disjunction, you should first be comfortable with logical statement. Once you have a solid grasp of disjunction, you can move on to and statement and truth table.
Visualization
StaticVisual representation of Disjunction