Question

An example of a relation on a set that is neither symmetric and anti symmetric.

Short answer

The example set is

$$\begin{gathered} R=\left\{\right(0,1),(1,2),(2,1\left)\right\} \\ R=\left\{\right(0,0),(1,5),(2,10),(10,2\left)\right\} \end{gathered}$$

Step-by-step solution

Given data

A relation on a set.

Concept used of relation

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

A relation role="math" localid="1668679211439" $\mathbf{\text{R}}$on a set role="math" localid="1668679222603" $\mathbf{\text{A}}$is called symmetric if role="math" localid="1668679234614" $\mathbf{(}\mathit{b}\mathbf{,}\mathit{a}\mathbf{)}\mathbf{\in }\mathit{R}$whenever role="math" localid="1668679243970" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$, for all role="math" localid="1668679255276" $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{A}$

A relation role="math" localid="1668679145395" $\mathbf{\text{R}}$on a set role="math" localid="1668679155003" $\mathbf{\text{A}}$such that for all role="math" localid="1668679166074" $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{A}$, if role="math" localid="1668679178367" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$and role="math" localid="1668679188391" $\mathbf{(}\mathit{b}\mathbf{,}\mathit{a}\mathbf{)}\mathbf{\in }\mathit{R}$then role="math" localid="1668679198835" $\mathit{a}\mathbf{=}\mathit{b}$is called anti symmetric.

A relation role="math" localid="1668679133119" $\mathbf{\text{R}}$on a set role="math" localid="1668679122363" $\mathbf{\text{A}}$is called transitive if whenever role="math" localid="1668679111081" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$and role="math" localid="1668679100428" $\mathbf{(}\mathit{b}\mathbf{,}\mathit{c}\mathbf{)}\mathbf{\in }\mathit{R}$then role="math" localid="1668679088960" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{c}\mathbf{)}\mathbf{\in }\mathit{R}$for all role="math" localid="1668679078111" $\mathit{a}\mathbf{,}\mathit{b}\mathbf{,}\mathit{c}\mathbf{\in }\mathit{A}$

Solve for relation

A relation is not symmetric nor anti symmetric if it contains an element $(a,b)$while it does not contain $(b,a)$and if it contains and element $(c,d)$while it also contains $(d,c)$with $c\ne d$.

For example:

$$\begin{gathered} R=\left\{\right(0,1),(1,2),(2,1\left)\right\} \\ R=\left\{\right(0,0),(1,5),(2,10),(10,2\left)\right\} \end{gathered}$$