I have an exam coming up soon and I cant find my notes on binary relations and have been unable to find anything on extending binary relations online. I would appreciate if someone could walk me through how to to this. I am not asking to do every question but one part would be nice so I can get a grasp on what I am doing. Thank you in advance.
Suppose two binary relations $R$ and $S$ over $(\mathbb{N}\times\mathbb{N})$ have been defined as:
$R=\{(2,1),(2,3),(3,2),(3,3),(4,1),(4,2)\}$
$S=\{(1,2),(2,3),(3,1)\}$
Write out each of the following binary relations in extension:
(a) $R;S$
(b) $S;R$
(c) $R\setminus S$
(d) $S;S;S$
(e) $R;(S)^{-1}$
(a) $R;S=\{(x,z): \exists y, xRy\land ySz\}$
So $R;S=\{(2,2),(2,1),(3,3),(3,1),(4,2),(4,3)\}$
(b) $S;R=\{(1,1),(1,3),(2,2),(2,3)\}$
(c) alreary answered (it is the set difference).
(d) $S;S;S=\{(1,1),(2,2),(3,3)\}$
(e) $S^{-1}=\{(2,1),(3,2),(1,3)\}$
So $R;(S)^{-1}=\{(2,3),(2,3),(3,2),(4,3),(4,1)\}$