Determine whether the relation $\ge$ is reflexive, symmetric, antisymmetric, transitive, and/or a partial order.

Question

Determine whether the relation is reflexive, symmetric, antisymmetric, transitive, and/or a partial order.

$ (x,y) \in R $ if $ x \ge y $ when defined on the set of positive integers.

I'm not sure how to even start this problem.

Answer 1

I’ll check two of the properties to give you the idea. I’m assuming that $R$ is a relation on the set of real numbers.

• Reflexivity: $R$ is reflexive if $\langle x,x\rangle\in R$ for every real number $x$. By the definition of $R$, $\langle x,x\rangle\in R$ if and only if $x\ge x$; is this true for every real number $x$? Definitely, so $R$ is reflexive.

• Symmetry: $R$ is symmetric if it has the following property: for any real numbers $x$ and $y$, if $\langle x,y\rangle\in R$, then $\langle y,x\rangle\in R$. For this specific relation that property says: for any real numbers $x$ and $y$, if $x\ge y$, then $y\ge x$. Is that true? Of course not: take $x=2$ and $y=1$, and we certainly have $x\ge y$, since $2\ge 1$, but it’s clearly not true that $y\ge x$, because $1\not\ge 2$. Thus, $R$ is not symmetric.

I’ll leave transitivity to you, just reminding you of the definitions.

• Antisymmetry: For any real numbers $x$ and $y$, if $\langle x,y\rangle\in R$ and $\langle y,x\rangle\in R$, then $x=y$. Is this true for this relation? Just translate $\langle x,y\rangle\in R$ and $\langle y,x\rangle\in R$ into more familiar terms, and it should be very clear.

• Transitivity: For any real numbers $x,y$, and $z$, if $\langle x,y\rangle\in R$ and $\langle y,z\rangle\in R$, then $\langle x,z\rangle\in R$. Again, if you translate the hypothesis into more familiar terms, you should have no trouble deciding whether the statement is true of this relation or not.