6.2.14 Binary relations

Problem: Two finite sets are given: \( A=\{a, b, c\}, B=\{1,2,3,4\} \); binary relations \( P_{1} \subseteq A \times B ; P_{2} \subseteq B^{2} \). Plot \( P_{1}, P_{2} \) graphically. Find \( P=\left(P_{2} \circ P_{1}\right)^{-1} \). Write the domain and the range of all three relation: \( P_{1}, P_{2}, P \). Construct the matrix \( \left[P_{2}\right] \), use it to check if the relation \( P_{2} \) is reflexive, symmetric, antisymmetric, transitive. \[ \begin{array}{l} P_{1}=\{(a, 2),(a, 4),(b, 1),(b, 2),(b, 4),(c, 2),(c, 4)\}, \\ P_{2}=\{(1,1),(2,2),(2,4),(3,3),(3,2),(4,4),(1,3),(4,1)\} . \end{array} \]