6.2.1 Binary relations

Problem: \( A=\{a, b, c\}, B=\{1,2,3,4\}, P_{1} \subseteq A \cdot B, P_{2} \subseteq B^{2} \). Represent \( P_{1}, P_{2} \) graphically. Find the matrix \( \left(P_{1} \circ P_{2}\right)^{-1} \). Using the matrix, check whether the relation \( P_{2} \) is reflexive, symmetric, antisymmetric, transitive? What class of relations does it belong to? \[ \begin{array}{l} P_{1}=\{(a, 1),(a, 2),(a, 3),(a, 4),(b, 3),(c, 2)\} \\ P_{2}=\{(1,1),(1,4),(2,2),(2,3),(3,3),(3,2),(4,1),(4,4)\} . \end{array} \]