Questionnaire-2024/smt/0014_sol.tex

 We choose: 
 \begin{itemize}
     \item Triangle 1: a-b-c
     \item Triangle 2: a-c-d
     \item Triangle 3: c-d-e
 \end{itemize}
 \begin{align*}
      \varphi_{TC} \coloneqq &
                       (e_{a = b} \land e_{b = c} \rightarrow e_{a = c}) \land \\
                       & (e_{a = b} \land e_{a = c} \rightarrow e_{b = c}) \land \\
                       & (e_{b = c} \land e_{a = c} \rightarrow e_{a = b}) \land \\
                       &\\
                       & (e_{a = c} \land e_{c = d} \rightarrow e_{a = d}) \land \\
                       & (e_{a = c} \land e_{a = d} \rightarrow e_{c = d}) \land \\
                       & (e_{c = d} \land e_{a = d} \rightarrow e_{a = c}) \land \\
                       &\\
                       & (e_{c = e} \land e_{c = d} \rightarrow e_{d = e}) \land \\
                       & (e_{c = e} \land e_{d = e} \rightarrow e_{c = d}) \land \\
                       & (e_{c = d} \land e_{d = e} \rightarrow e_{c = e}) 
        \end{align*}
    \begin{align*}
      \hat\varphi_{E} \coloneqq
                          (e_{a=b} \; \lor \; e_{a=d} \imp (e_{b=c} \; \land \; \neg e_{c = e}\; \land \; \neg e_{e = d})
    \end{align*}
    \begin{align*}
     \varphi_{prop} \coloneqq \varphi_{TC} \land \hat\varphi_{E} 
    \end{align*}