Let $V = \left\{1,...,5\right\}$ and there is relation $R$ on $V$
$$R=\left\{(1,1),(1,5),(2,4),(3,3),(4,1),(4,2),(5,4)\right\}$$
What is $R^T, R^2, R^+, h_{\text{sym}}(R)$?
Hi maths people I need info how these notation is correct for learn it because not sure I understand all notation good.
I check on internet $R^T$ mean transpose of relation $R$ is defined: $R^T= \left\{(y,x) | xRy\right\}$
That's why I write $R^T=\left\{(1,1),(5,1),(4,2),(3,3),(1,4),(2,4),(4,5)\right\}$
$R^2$ I don't know..
Definition of $R^+ = \bigcap\left\{S \subseteq A \times A \mid S \text{ is transitive and } R \subseteq S\right\}$
I understand that I need take all pair from $R$ which keep it transitive?
$R^+ = \left\{(1,1), (5,1)\right\}$
Definition $h_{\text{sym}}(R)=\bigcap\left\{S \subseteq A \times A \mid S \text{ is symmetric and } R \subseteq S\right\}$
I take all pair from $R$ as long as it's symmetric?
$h_{\text{sym}}(R)= \left\{(1,1), (3,3), (2,4),(4,2)\right\}$
Please help for understand it I do it good or not?
$R^2$ = { (a,b) : some x with aRx, xRb }
For $R+$ add as few pairs as possible to R to make R transitive.
For sym(R) add as few pairs as possible to R to make R symmetric.