Binary relation

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A subset of LaTeX: S \times T, say LaTeX: R. When LaTeX: T=S, we refer to just the relation on the set LaTeX: S.

Here are 5 properties that distinguish relations on a set (LaTeX: x, LaTeX: y, LaTeX: z are in LaTeX: S):

  1. antisymmetric - LaTeX: (x, y) \in R and LaTeX: x \ne y imply LaTeX: (y, x) \not\in R.
  2. irreflexive - LaTeX: (x, x) \not\in R.
  3. reflexive - LaTeX: (x, x) \in R.
  4. symmetric - LaTeX: (x, y) \in R implies LaTeX: (y, x) \in R.
  5. transitive - LaTeX: (x, y) and LaTeX: (y, z) both in LaTeX: R imply LaTeX: (x, z) \in R.

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