|Category: utilities||Component type: concept|
|X||A type that is a model of LessThanComparable|
|x, y, z||Object of type X|
If operator< is a strict weak ordering, and if each equivalence class has only a single element, then operator< is a total ordering.
|Name||Expression||Type requirements||Return type|
|Less||x < y||Convertible to bool|
|Greater||x > y||Convertible to bool|
|Less or equal||x <= y||Convertible to bool|
|Greater or equal||x >= y||Convertible to bool|
|Less||x < y||x and y are in the domain of <|
|Greater||x > y||x and y are in the domain of <||Equivalent to y < x |
|Less or equal||x <= y||x and y are in the domain of <||Equivalent to !(y < x) |
|Greater or equal||x >= y||x and y are in the domain of <||Equivalent to !(x < y) |
|Irreflexivity||x < x must be false.|
|Antisymmetry||x < y implies !(y < x) |
|Transitivity||x < y and y < z implies x < z |
 Only operator< is fundamental; the other inequality operators are essentially syntactic sugar.
 Antisymmetry is a theorem, not an axiom: it follows from irreflexivity and transitivity.
 Because of irreflexivity and transitivity, operator< always satisfies the definition of a partial ordering. The definition of a strict weak ordering is stricter, and the definition of a total ordering is stricter still.
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