Question

Determine the following relation is an equivalence relation or not. If the relation is an equivalence relation, describe the partition given by it. xRy if x>yx,y∈the set of all real numbers

Answer 1

The relation is given by

xRy if x>y , x,y $\in$ $\mathbb {R}$ , set of reals

i.e. R = { (x,y) : x > y , x,y $\isin \mathbb {R}$ }

To show equivalency of the relation..

$\mathbf {REFLEXIVITY}$ :

For a real x, x $\ngtr$ x

So (x,x) $\notin$ R

Therefore R is not reflexive

$\mathbf {SYMMETRY}$ :

Let (x,y) $\isin$ R for x,y $\in \mathbb {R}$

So x > y

Obviously y $\ngtr$ x

Therefore (y,x) $\notin$ R

So R is not symmetric.

$\mathbf {TRANSITIVITY}$ :

Let (x,y) $\in$ R and (y,z) $\isin$ R for x,y,z $\in \mathbb {R}$

So x > y and y > z

Obviously x > z

So R is transitive.

A relation be equivalence relation if it is reflexive, symmetric and transitive.

In this case R is transitive but neither reflexive nor symmetric.

So given R is not an equivalence relation.

As it is not equivalence relation , second part of the question is not applicable.