Question

To determine whether the relation$\mathit{R}$ on the set of all real numbers is reflexive, symmetric, anti symmetric, transitive, where $\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{)}\mathbf{\in }\mathit{R}$ if and only if$\mathit{x}\mathbf{\ge }{\mathit{y}}^{\mathbf{2}}$.

Short answer

The given set is anti symmetric.

Step-by-step solution

Given data

$x\ge y^2$is the given condition and$(x,y)\in R$.

Concept used of relation

A relation$\mathbf{\text{R}}$on a set$\mathbf{\text{A}}$ is called reflexive if$\mathbf{(}\mathit{a}\mathbf{,}\mathit{a}\mathbf{)}\mathbf{\in }\mathit{R}$ for every element$\mathit{a}\mathbf{\in }\mathit{A}$.

A relation $\mathbf{\text{R}}$on a set is called symmetric if$\mathbf{(}\mathit{b}\mathbf{,}\mathit{a}\mathbf{)}\mathbf{\in }\mathit{R}$whenever$\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$ for all $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{A}$.

A relation$\mathbf{\text{R}}$ on a set $\mathbf{\text{A}}$such that for all$\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{A}$, if $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$and $(\mathit{b}\mathbf{,}\mathit{a}\mathbf{)}\mathbf{\in }\mathit{R}$then $\mathit{a}\mathbf{=}\mathit{b}$ is called anti symmetric.

A relation$\mathbf{\text{R}}$on a set$\mathbf{\text{A}}$is called transitive if whenever$\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$and$\mathbf{(}\mathit{b}\mathbf{,}\mathit{c}\mathbf{)}\mathbf{\in }\mathit{R}$thenfor all$\mathit{a}\mathbf{,}\mathit{b}\mathbf{,}\mathit{c}\mathbf{\in }\mathit{A}$

Solve for relation

Not reflexive because we can't have$(2,2).$

Not symmetric because if we have$(9,3)$, we can't have$(3,9)$.

Is anti symmetric, because each integer will map to another integer but not in reverse (besides 0 and 1 ).

Is transitive because if $x\ge y^2$and $y\ge z^2$ then $x\ge z^2$.

The given set is anti symmetric.