Answer to Question #117142 in Discrete Mathematics for Priya

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.