MathProblemsBank

6.2.18 Binary relations

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