Partial Order Relations with irreflexive definitions

Question

Define a relation R on the set of real numbers by (x,y) R if and only if x - y = 0. Determine if the relation R is a partial order. If it is not a partial order, explain which property or properties it fails to have.

Since for all x=y, x-y=0, this relation would be irreflexive, correct? I am not sure...

Answer 1

You’ve just shown that $\langle x,y\rangle\in R$ if and only if $x=y$. In other words, $R$ is just the relation of equality on $\Bbb R$. This is a standard example of a relation that is reflexive: for each $x\in\Bbb R$, $x=x$, so $\langle x,x\rangle\in R$.

The other two properties that need to be checked are antisymmetry and transitivity.

• Antisymmetry: Is it true that if $\langle x,y\rangle\in R$ and $\langle y,x\rangle\in R$, then $x=y$?
• Transitivity: Is it true that if $\langle x,y\rangle\in R$ and $\langle y,z\rangle\in R$, then $\langle x,z\rangle\in R$?

In both cases you should begin by translating statments like $\langle x,y\rangle\in R$ into more basic statements about $x$ and $y$ by using the definition of $R$. For instance, we’ve already seen that $\langle x,y\rangle\in R$ means simply that $x=y$.

Answer 2

The relation $R$ is reflexive if for any real number $x$ it is the case that $(x,x) \in R$. Is it the case that for any real number you have $x - x = 0$?