Relation

Definition:

• A binary relation, R, consists of a set A, called domain of R, a set B, called codomain of R, and a subset of graph of $R$​
  • We write $R:A\to B$ to indicate that R is a relation from A to B
  • “$aRb$: the pair $(a,b)$ is in the graph of R, means $a$ and $b$ belongs to $R$
  • For every ordered tuple pair, $(x,y)\in R,x\in A,y\in B$ and is subset of $A\times B$

Graph of $R$:

• $A\times B$: Cartesian product of domain, set $A$, and codomain, set $B$.

Types of relation:

1. Reflexive:
   • if $(x,x)\in R$ for all $x\in S$
     • 10 mod 3 = 10 mod 3
     • every point has a path to itself
2. Symmetric:
   • if whenever $(x,y)\in R$ then $(y,x)\in R$
     • x → y and y → x
     • $xRy\to yRx$
     • Every path has a reverse path​
3. Transitive:
   • whenever $(x,y)\in R,(y,z)\in R$ and $(x,z)\in R$
     • • x → y → z and x → z
     • every 2-step path has a 1-step path

Function

Inverse relation:

• The inverse, $R^{-1}$ of a relation $R:A\to B$ is the relation from $B$ to $A$ defined by the rule $b{R}^{-1}a⟺aRb$