composition of binary relation with condition

Question

I have the binary relation :

$R = \{(x,y) | x,y \in N, xy <5 \}$

I need to find the composition of R - $R \circ R $

Then given answer is : $N * N$

which it does not makes sense, can you explain it please?

Answer 1

We have that $R \subseteq \mathbb N \times \mathbb N$ and $R = \{ (x,y) \mid x,y \in \mathbb N \text { and } xy < 5 \}$.

Simple examples: $(1,4) \in R$ and $(2,3) \notin R$.

Now, for composition of $R$, i.e $R \circ R$ :

$(x,z) \in R \circ R \text { iff } \exists y \in \mathbb N : (x,y) \in R \text{ and } (y,z) \in R$.

But this amount to saying that $(x,z) \in R \circ R$ iff $xy<5$ and $yz <5$ for some $y \in \mathbb N$.

IF $0 \in \mathbb N$, we have that $x0 <5$ and $0z < 5$ for every $x,z \in \mathbb N$.

Thus:

$R \circ R = \mathbb N \times \mathbb N$.