Finding equivalence class with a binary set

Question

I'm new to discrete math so there might be problems with this solution. Prompt is to find at least one equivalence class if it is an equivalence relation.

$$X = R^2, (x_1, y_1) \sim (x_2, y_2) \iff y_1 = y_2$$

1. Reflexivity

$$(x_1, y_1) \sim (x_1, y_1) \iff y_1 = y_1$$

Since $y_1 = y_1$, it is reflexive.

2. Symmetry

$$(x_1, y_1) \sim (x_2, y_2) \iff y_1 = y_2$$

Assuming $y_1 = y_2$ (eqn1),

$$(x_2, y_2) \sim (x_1, y_1) \iff y_2 = y_1$$

from eqn 1, $y_2 = y_1$, so it is a symmetry.

3. Transitivity

$$(x_1, y_1) \sim (x_2, y_2) \iff y_1 = y_2$$

$$(x_2, y_2) \sim (x_3, y_3) \iff y_2 = y_2$$

Assuming $y_1 = y_2$ and $y_2 = y_3$, we get $y_1 = y_3$.

$$(x_1, y_1) \sim (x_3, y_3) \iff y_1 = y_3,$$

since $y_1 = y_3$, so it is transitive.

Is this the correct way to solve the problem? Also how to find the equivalence classes for the same?

Answer 1

You did correctly. Well.. to find the equivalent class, you often need a representant of that class. Take $(x, y)$, and we want $[(x, y)]$ to be an equivalent class which contains the element $(x, y)$. In other words, $(x, y)$ represents that class. That is: $$ [(x, y)] = \{(a, b)\in\mathbb{R}^2 : (x, y)\sim(a, b)\} $$

We have an equivalent class such that $(x, y)\in [(x, y)]$. Now, its all a matter of inserting the definition of $\sim$ in the set. $$ [(x, y)] = \{(a, b)\in\mathbb{R}^2 : y=b\} $$

Can you come up with a concrete example? What would $[(1, 2)]$ be, for instance?

Answer 2

Yes, your proof is correct. To construct an equivalence class, pick an element of the set and find (describe) all elements that are equivalent to it. More formally: if $R$ is an equivalence relation on a set $X$, then for an $x\in X$, its equivalence class is $[x]=\{y\in X\,\colon\,yRx\}$. In this example: pick an arbitrary $(x,y)\in\mathbb{R}^2$. By the given definition, any other $(x_1,y_1)\in\mathbb{R}^2$ is equivalent to it, $(x_1,y_1)\sim(x,y)$, iff $y_1=y$, so $(x_1,y_1)=(x_1,y)$. Note that there are no constraints on $x_1$, so it can be any real. Thus the equivalence class of $(x,y)$ is $\{(x_1,y)\in\mathbb{R}^2\,\colon\,x_1\in\mathbb{R}\}$ (where $y$ is fixed). Geometrically, they are horizontal lines $y=\text{const}$.