[Discrete Mathematics] Relations

1. Introduction

a. Definition

  • Let $A$ and $B$ be sets. A binary relation from $A$ to $B$ is a subset of A x B.$$ R \subseteq A × B $$$$ aRb : (a,b) \in R $$
  • relation on a set A is a relation from A to A.

b. Properties of relations

  • A relation $R$ on a set $A$ is called reflexive if $ (a, a) \in R, \forall a \in A $
  • A relation $R$ on a set $A$ is called symmetric if $ (a, b) \in R \Rightarrow (b,a) \in R, \forall a, b \in A $
  • A relation $R$ on a set $A$ is called antisymetric” if $ aRb \land bRa \Rightarrow a=b, \forall a,b \in A $
  • A relation $R$ on a set $A$ is called transitive if $ aRb \land bRc \Rightarrow aRc, \forall a,b,c \in A $

References

  • Discrete Mathematics and Its Applications 7th Edition, Kenneth H. Rosen
  • Đại số tuyến tính, Nguyễn Hữu Việt Hưng
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