## Non-strict orders: ≤

The symbol ≤ denotes a generalization of “less than or equal”, and it defines either a partial or total ordering over a set P (in the table below a,b ∈ P):

Constraint | (Non-strict) partial order | (Non-strict) total order |
---|---|---|

Reflexivity: a ≤ a |
x | x |

Antisymmetry: if a ≤ b and b ≤ a then a = b |
x | x |

Transitivity: if a ≤ b and b ≤ c then a ≤ c |
x | x |

Totality: either a ≤ b or b ≤ a |
x |

## Strict orders: <

The symbol < denotes a generalization of “less than”, and it defines either a partial or total ordering over a set P (in the table below a,b ∈ P):

Constraint | (Strict) partial order | (Strict) total order |
---|---|---|

Irreflexivity: ¬(a < a) |
x | x |

Asymmetry: if a < b then ¬(b < a) |
x | x |

Transitivity: if a < b and b < c then a < c |
x | x |

Totality: either a < b or b < a |
x |

Note the difference between asymmetry and antisymmetry.

Type of relation | Constraint |
---|---|

Asymmetric relation | if a < b then ¬(b < a) |

Antisymmetric relation (two equivalent definitions) |
if a ≤ b and b ≤ a then a = b |

if a ≠ b then ¬(b ≤ a) |