# discrete 7.1 relation

## relation
R is relation on A
R $\subseteq$ AxA, if a,b$\in$A,(a,b)$\in$R, called aRb
$R_1*R_2$ = $R_1。R_2$ is like f(g(x))

$R*R = R^2$ is like $f(f(x))$

$R = {(1,1),(1,2),(2,3),(3,3),(3,4)}$

$RR = {(1,f(1)),(1,f(2))....}$
$RR = {(1,1),(1,2),(1,3),(2,3),(2,4),(3,3),(3,4)}$
[explain](https://youtu.be/aEcAh-lcsgY)

## reflexive 反射性
set A, x$\in$A (x,x)$\in R$
$\forall x \in Z, x \le x$Rexflexive

(1,1),(2,2),(3,3)
## symmetric 對稱性
if (𝑥, 𝑦)∈ℜ⟹(𝑦, 𝑥)∈ℜ for all 𝑥, 𝑦∈

𝑎,𝑏∈𝐴,[(𝑎, 𝑏)or(𝑏,𝑎) $\notin$ ℜ]⟹𝑎$\ne$𝑏.
(1,2),(2,1),(1,3),(3,1)

(1,1)(2,2),(3,3)
1 x

## transitive 遞移性
all 𝑥, 𝑦,𝑧∈𝐴, (𝑥, 𝑦),(𝑦,𝑧)∈ℜ⟹(𝑥, 𝑧)∈ℜ.

(1,2),(2,3),(1,3)
or
1<2, 2<3,可推 1<3,so3$\nless$

operator x-y = even
(2,4)(4,6)()

## antisymetric 反對稱性
antisymmetric if, for all 𝑎,𝑏∈𝐴,[(𝑎, 𝑏),(𝑏,𝑎)∈ℜ]⟹𝑎$\ne$𝑏.

(1,1)(2,3)(3,3)
(1,1)(2,2),(3,3)

operator $\le$

## partial ordering
if $R$ have reflexive, transitive, antisymetric

$(a,b) \in R, a\le b$

# reference

[relation graph](https://math24.net/properties-relations.html)