6.2.13 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 range of all three relations: \( P_{1}, P_{2}, P \). Construct the matrix \( \left[P_{2}\right] \), use it to check whether the relation \( P_{2} \) is reflexive, symmetric, antisymmetric, transitive. \[ \begin{array}{l} P_{1}=\{(a, 3),(b, 4),(b, 3),(c, 1),(c, 2),(c, 4)\} ; \\ P_{2}=\{(1,2),(1,3),(1,4),(2,3),(4,3),(4,2)\} . \\ A=\{a, b, c\}, B=\{1,2,3,4\}, P_{1} \subseteq A \times B, P_{2} \subseteq B^{2} . \end{array} \]