Expressing Relations with Quantifiers

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A \emph{reflexive} relation is one that everything bears to itself. (E.g. SameShape)

reflexive: ∀x R(x,x)

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A \emph{symmetric} relation is one such that if x bears it to y, then y bears it to x. (E.g. Adjacent(x,y))

symmetric: ∀x∀y ( R(x,y) → R(y,x) )

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A \emph{transitive} relation is one such that if x bears it to y and y bears it to z then x bears it to z. (E.g. LeftOf is transitive; DifferentShape is not transitive)

transitive: ∀x∀y∀z ( ( R(x,y) ∧ R(y,z) ) → R(x,z) )

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