Calculating a union of 2 relations

Question

I have 2 relations: $$ xSy \Leftrightarrow y = 2x$$ and $$ xTy \Leftrightarrow y = 3x$$ The problem I have is calculating $$x(T \cup S)y$$ and $$xS^+y $$

Could you please help me?

Answer 1

Start with an arbitrary $$x\in \mathbb{N}$$ then $$(x,2x)\in S, (x,4x)\in S^2, (x,8x)\in S^3, \ldots, (x,2^nx)\in S^n, \ldots$$ so we can say

$$ xS^+y \Leftrightarrow \exists n\in \mathbb{N} s.t. y=2^nx $$