# Exhaustive and exclusive alternatives

## Definitions

Here we are taking different alternatives to be described by statements.

- A set of statements is
*exhaustive* when at least one of them is true in any logically possible situation. ("exhaustive" because they do not leave out any situation)
- A set of statements is
*exclusive* when there is no logically possible situation in which more than one of them is true. ("exclusive" because the truth of one excludes the truth of the others)
- In other words, if a set of statements is both exhaustive and exclusive, then in any logically possible situation, exactly one of them is true.

## Examples

- Neither exhaustive nor exclusive - a=5, a>4
- Exhaustive but not exclusive - a>4, a<10
- Exclusive but not exhaustive - a>4, a=1
- Exclusive and exhaustive - a>0, a=0, a<0

## Exercises

- "The market will go up. The market will go down." Why are they not exhaustive alternatives?
- "P and Q", "Neither P nor Q" - Are they exclusive? exhaustive?
- "John is happy", "John is sad" - Are they exclusive? exhaustive?

Category.LogicAndMaths