Q1: A relation $S$ on $\mathbb{N}$ is given by $xSy$ if $x=2^{\alpha}y$ for some $\alpha \in \mathbb{Z}$.
Find the equivalence class for the element 2.
Find the set of equivalence classes.
Q2: A relation $T$ on $\mathbb{N} \times \mathbb{N}$ is given by $(a,b)T(c,d)$ if $a+d=b+c$.
- Find the equivalence class for the element $(2,0)$.
- Find the set of equivalence classes.
What I've done for Q1:
$2=2^{\alpha}y$ so $y=2^{1-\alpha}$ therefore the equivalence class is $[2]_S=\{2^{1-\alpha}|\alpha \in \mathbb{Z}\}$ Then for the set of all classes I've repeated above for some arbitrary $x$ in $\mathbb{N}$. So $\mathbb{N}/S= \{ x*2^{-\alpha}|\alpha \in \mathbb{Z} \}$.
Is this correct?
What I've done for Q2:
When we have $(2,0)$, $a+d=b+c$ gives $c-d=2$ so $[(2,0)]_T=\{c-d=2|c,d \in \mathbb{N} \}$ however, for the set of classes $\mathbb{N}/T$ I'm not sure how to move forward. $a+d=b+c$ can be rearranged into $a-b=c-d$ which might help as you have $a,b$ on the left and $c,d$ on the right. But don't know where to go from here.