---
tag: 0H5GKDTB01
kind: definition
name: Disjunction
msc: [03B05]
defines: 0F72FK8EKB
uses: [0ZGC7ZESYN]
sources:
  - url: https://en.wikipedia.org/wiki/Logical_disjunction
    locator: lead section
---

## Definition

Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]].
The disjunction of $p$ and $q$ is the proposition $p \lor q$, read "$p$ or $q$".
The proposition $p \lor q$ is false when $p$ is false and $q$ is false. In all other cases, $p \lor q$ is true.

| $p$ | $q$ | $p \lor q$ |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |

## Remarks

This "or" is inclusive: $p \lor q$ is true when both $p$ and $q$ are true.
The exclusive "or" is true when exactly one of $p$ and $q$ is true.
The [[union|07KVB4JSC7]] of two sets uses disjunction: $x \in A \cup B$ means $x \in A \lor x \in B$.

## Explanations

### L1 human

The sentence "$4$ is even or $4$ is odd" is a disjunction.
It is true, because one part is true.
In mathematics, "or" also allows both parts to be true.