Irreflexive version
A set X is partially ordered by R if:
- For elements x of X, we do not have xRx (R is irreflexive)
- We cannot have both xRy and yRx (R is anti-symmetric)
- xRy and yRz implies xRz (R is transitive)
This is intuitively like "strictly less than" - $<$
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Reflexive version
A set X is partially ordered by R if:
- For elements x of X, we do have xRx (R is reflexive)
- If xRy and yRx then x=y (R is anti-symmetric)
- xRy and yRz implies xRz (R is transitive)
This is intuitively like "less than or equal to" - $\le$
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