If $R = \{(1,2),(1,4),(3,3),(4,1)\}$, then is $(1,2) \in R^2$? (Powers of Relation)

Question

I basically got this:

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

But I'm not sure if I should include (1,2) as well since 2 maps to nothing?

Thanks

Answer 1

Rather, $R$ maps $1$ to $2$, and maps $1$ to $4$, but does not map either $2$ or $4$ to $2$, so $R^2$ does not map $1$ to $2$.

$$\big(\neg\exists y~.(\langle 1,y\rangle\in R\wedge\langle y,2\rangle\in R)\big)~\to~ \langle 1,2\rangle\notin R^2$$

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$${R^2 =\{\langle x, z\rangle: \exists y~.(\langle x,y\rangle\in R\wedge\langle y,z\rangle\in R)\}\\\quad=\{\langle 1,1\rangle, \langle 3,3 \rangle, \langle 4,2 \rangle, \langle 4, 4\rangle\}}$$