Equivalence Classes points on the plane

Question

I'm confused about the topic of equivalence classes. $x=(a,b) , y=(c,d)$ are points on the plane. $xRy$ iff:

1) $a+b = c+d$

2) $a^2-b = c^2-d$

3) $a=c=5 , b=d=20$

4) $a^4+b^4 = c^4+d^4$

For each one of these, what are the equivalence classes??

Answer 1

Hint:

Equivalence class of an element $a \in A$ for an equivalence relation $R: A \longrightarrow A$ is defined as the set of those elements in $A$ which are ''equivalent'' to $a$ via relation $R$.

In your case the set $A=\mathbb{R}^2=\{(x,y) \, | \, x,y \in \mathbb{R}\}$.

For the relation in part (1).

Consider the equivalence class of the element $(s,t) \in A$. By definition $$[(s,t)]_{R}=\{(x,y) \in A \, | \, s+t=x+y\}.$$ But what exactly is this set? Here is an example to help you understand what this set actually represents.

Suppose we want the equivalence class of the point $(2,3) \in A$. Then \begin{align*} [(2,3)]_{R} & =\{(x,y) \in A \, | \, 2+3=x+y\}\\ & =\{(x,y) \in A \, | \, x+y=5\}. \end{align*} This basically is the set of all points lying on the straight line $x+y=5$. Observe that this is a straight line with slope $-1$ and passes through the point $(2,3)$.

So now you can generalize this to say that $[(s,t)]_R$ is the set of all points which lie on the straight line with slope $-1$ and passing through the point $(s,t)$.

Similarly try the other parts.