# Conjunction elimination

Type | Rule of inference |
---|---|

Field | Propositional calculus |

Statement | If the conjunction [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] is true, then [math]\displaystyle{ A }[/math] is true, and [math]\displaystyle{ B }[/math] is true. |

Transformation rules |
---|

Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

In propositional logic, **conjunction elimination** (also called **and****elimination**, **∧ elimination**,^{[1]} or **simplification**)^{[2]}^{[3]}^{[4]} is a valid immediate inference, argument form and rule of inference which makes the inference that, if the conjunction *A and B* is true, then *A* is true, and *B* is true. The rule makes it possible to shorten longer proofs by deriving one of the conjuncts of a conjunction on a line by itself.

An example in English:

- It's raining and it's pouring.
- Therefore it's raining.

The rule consists of two separate sub-rules, which can be expressed in formal language as:

- [math]\displaystyle{ \frac{P \land Q}{\therefore P} }[/math]

and

- [math]\displaystyle{ \frac{P \land Q}{\therefore Q} }[/math]

The two sub-rules together mean that, whenever an instance of "[math]\displaystyle{ P \land Q }[/math]" appears on a line of a proof, either "[math]\displaystyle{ P }[/math]" or "[math]\displaystyle{ Q }[/math]" can be placed on a subsequent line by itself. The above example in English is an application of the first sub-rule.

## Formal notation

The *conjunction elimination* sub-rules may be written in sequent notation:

- [math]\displaystyle{ (P \land Q) \vdash P }[/math]

and

- [math]\displaystyle{ (P \land Q) \vdash Q }[/math]

where [math]\displaystyle{ \vdash }[/math] is a metalogical symbol meaning that [math]\displaystyle{ P }[/math] is a syntactic consequence of [math]\displaystyle{ P \land Q }[/math] and [math]\displaystyle{ Q }[/math] is also a syntactic consequence of [math]\displaystyle{ P \land Q }[/math] in logical system;

and expressed as truth-functional tautologies or theorems of propositional logic:

- [math]\displaystyle{ (P \land Q) \to P }[/math]

and

- [math]\displaystyle{ (P \land Q) \to Q }[/math]

where [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q }[/math] are propositions expressed in some formal system.

## References

sv:Matematiskt uttryck#Förenkling

Original source: https://en.wikipedia.org/wiki/Conjunction elimination.
Read more |