# Propositional Equivalences

## Types of Propositional Equivalences

- Tautologies
- Contradictions
- Contigency
- Logical equivalences

### Tautologies

A tautology is a statement that is always true.

Examples:

- R∨(¬R)
- ¬(R∧Q) ↔ (¬P) ∨ (¬Q)
- Excercise: Check that the above statements are always true
- If S → T is a tautology, we write S ⇒ T
- If S ↔ T is a tautology, we write S ⇔ T
### Tautologies and Contradictions

The negation of any tautology is a contradiction, and the negation of any contradiction is a tautology.

#### Exercise: Check if the following contradictions are always false:

- R ∧ (¬R)
- ¬(¬(P∧Q)↔(¬P)∨(¬Q))

### Contigency

A contigency is a proposition that is neither a contradiction nor a tautology.

Example: R ∨ S → ¬T

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Date of last modification: February 12, 2019