##Unscramble the following definitions. Let $A$ and $B$ be sets.

#$R$ is a relation from $A$ to $B$

$\Leftrightarrow$ $R\subseteq A\times B$

#$R$ is a binary relation on $A$

$\Leftrightarrow$ $R\subseteq A\times A$

##Let $R$ and $S$ be binary relations on $A$.

#$R$ is reflexive

$\Leftrightarrow$ $\forall x\in A\ (x,x)\in R$

#$R$ is irreflexive

$\Leftrightarrow$ $\forall x\in A\ (x,x)\notin R$

#$R$ is symmetric

$\Leftrightarrow$ $\forall x,y\ [(x,y)\in R\to (y,x)\in R]$

#$R$ is antisymmetric

$\Leftrightarrow$ $\forall x,y\ [(x,y)\in R\wedge (y,x)\in R\to x=y]$

#$R$ is asymmetric

$\Leftrightarrow$ $\forall x,y\ [(x,y)\in R\to (y,x)\notin R]$

#$R$ is transitive

$\Leftrightarrow$ $\forall x,y,z\ [(x,y)\in R\wedge (y,z)\in R\to
(x,z)\in R]$

#$R$ composed with $S=S\circ R$

$=\{(x,y)\ |\ \exists y\ [(x,y)\in R\wedge (y,z)\in S]\}$

#$(x,z)\in S\circ R$

$\Leftrightarrow$ $\exists y\ [(x,y)\in R\wedge (y,z)\in S]$

#$R^{-1}$

$=\{(y,x)\ |\ (x,y)\in R\}$

#For $n>0$, $R^n$ is defined by

$R^1=R$ and $R^{n+1}=R^n\circ R$

#$(x,y)\in R^n$

$\Leftrightarrow$ $\exists x_0,x_1,\ldots,x_n\ [x_0=x\wedge x_n=y\wedge
(x_i,x_{i+1})\in R$ for $i<n]$