Let R be a binary relation on a set S. Then R is called: (R is a set or ordered tuple)
reflexive(R): iff (\forall x \in S)((x,x) \in R) (any element is related to itself) 自己和自己比永远存在
irreflecsive(IR): iff (\forall x \in S)((x,x) \notin R) (no element is related to itself, notice it is not the opposite of relfecsive) 自己和自己比永远不存在
symmetric(S): iff (\forall x,y \in S)((x, y) \in R \implies (y, x) \in R) 正反永远成立
antisymmetric(AS): iff (\forall x,y \in S)((x, y) \in R \land (y, x) \in R \implies x = y) 正反成立则相等
transitive(TR): iff (\forall x, y, z \in R)((x, y)\in R \land (y, z)\in R \implies (x, z) \in R) 传递
Total(linear)(To): (\forall x, y \in S)(x \neq y \implies ((x, y) \in R \lor (y, x) \in R)) 两个数总有联系
Partial Order: iff R is reflexive, antisymmetric, and transitive. (S, R) poset. Strict Partial Order: iff R is irefflexive, antisymmetric, and transitive. (S, R) strict poset.
Equivalence: reflexive, symmetric, and transitive.
S/R = \{[x]_R | x \in S\} \mathbb{Z}/m\mathbb{Z} is for congruent modulo m
Definition: Left S be a nonempty set
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