### Example 6.3.

Let \(X=\{a,b,c,d,e,f\}\text{.}\) Consider the following binary relations on \(X\text{.}\)

\begin{align*}
R_1=\{\amp (a,a),(b,b),(c,c),(d,d),(e,e),(f,f),(a,b),(a,c),(e,f)\}\\
R_2=\{\amp (a,a),(b,b),(c,c),(d,d),(e,e),(f,f),(d,b),(d,e),(b,a),(e,a),\\
\amp(d,a),(c,f)\}\\
R_3=\{\amp(a,a),(b,b),(c,c),(d,d),(e,e),(f,f),(a,c),(a,e),(a,f),(b,c),\\
\amp(b,d),(b,e),(b,f),(d,e),(d,f),(e,f)\}\\
R_4=\{\amp(a,a),(b,b),(c,c),(d,d),(e,e),(f,f),(d,b),(b,a),(e,a),(c,f)\}\\
R_5=\{\amp(a,a),(c,c),(d,d),(e,e),(a,e),(c,a),(c,e),(d,e)\}\\
R_6=\{\amp(a,a),(b,b),(c,c),(d,d),(e,e),(f,f),(d,f),(b,e),(c,a),(e,b)\}
\end{align*}

Which of the binary relations are partial orders on \(X\text{?}\) For those that are not partial orders on \(X\text{,}\) which property or properties are violated?

Solution.

A bit of checking confirms that \(R_1\text{,}\) \(R_2\) and \(R_3\) are partial orders on \(X\text{,}\) so \(\bfP_1=(X,R_1)\text{,}\) \(\bfP_2=(X,R_2)\) and \(\bfP_3=(X,R_3)\) are posets. Several of the other examples we will discuss in this chapter will use the poset \(\bfP_3=(X,R_3)\text{.}\)

On the other hand, \(R_4\text{,}\) \(R_5\) and \(R_6\) are not partial orders on \(X\text{.}\) Note that \(R_4\) is not transitive, as it contains \((d,b)\) and \((b,a)\) but not \((d,a)\text{.}\) The relation \(R_5\) is not reflexive, since it doesn’t contain \((b,b)\text{.}\) (Also, it also doesn’t contain \((f,f)\text{,}\) but one shortcoming is enough.) Note that \(R_5\) is a partial order on \(\{a,b,d,e\}\text{.}\) The relation \(R_6\) is not antisymmetric, as it contains both \((b,e)\) and \((e,b)\text{.}\)