How would I show that this is a subset?

Question

Let R1 and R2 be relations on N defined by xR1y if and only if y=a+x for some a ∈ N0. xR2y if and only if y=xa for some a ∈ N. for all x,y ∈ N. Also N0 denotes all integers x>=0, while N denotes all integers >= 1.

I want to write a proof that shows that R2⊆R1.

I'm not sure how to write proofs and I need someone to help me please as this is one of the practice problems to help study for my final exam, so I want to be able to write good proof's

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I have another part if any of you could help.

I was already able to write a proof for how R1 is a partial order, but now I want to write one to show that R1 is a total order. Since I already proved that it is anti-symmetric and transitive I just need to prove it is a totality but I don't know how to do this, can anyone help?

Answer 1

Without words

$$(x,y)\in R_2\implies \exists\,a\in\Bbb N\;\;s.t.\;\;y=xa=x+(a-1)x\implies (x,y)\in R_1$$

Answer 2

Let $(x,y)\in R_2$ so it will exist $a\in N$ such that $y=xa=x(1+a-1)=x+x(a-1)$. $x(a-1) \in N_0$ so $(x,y)\in R_1$