relations

relations - discrete math

Relation description

Def 1:

• a binary relation R consists
  • set A: domain of R
  • set B: codomain of R
  • subset of A x B: graph of R
• example:
  • A = {1,2}
  • B = {a,b,c},
  • R = {(1,a),(1,b),(2,c)} is a relation from A to B, or a subset of A x B
• note:
  • $R:A->B$: R is a relation from A to B
  • "$aRb$": the pair (a, b) in the graph of R
• confusing example:
  • eg1

Relation Types:

Reflexive, Symmetric, Transitive

• A relation R on set S is:
• Reflexive:
  • every element in S is related to itself
  • if $(x,x)\in R$ for all $x\in S$
  • eg: the “equal” relation is reflexive, as every number is equal to itself
  • tips: is x “the relation” to x correct
• Symmetric:
  • order of elements in the pair don’t matter
  • $(x,y)\in R$, then $(y,x)\in R$
  • eg. the “is sibling of relation is symmetric” as if x is sibling of y, then y is also sibling of x
• Transitive: (suy ra)
  • if x related to y, y related to z, then x is also related to z
  • if $(x,y)\in R$, and $(y,z)\in R$, then $(x,z)\in R$

Inverse relation

• $R^{-1}$ of $R:A\to B$ is the relation from B to A defined by $(b{R}^{-1}a)IFF(aRb)$
• change the position & reverse arrow sign: $A\to B$ into $B\to A$