
6.2.16 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 of definition and ranges of all three relations: \( P_{1}, P_{2}, P \). Construct the matrix \( \left[P_{2}\right] \), use it to check if the relation \( P_{2} \) is reflexive, symmetric, antisymmetric, transitive.
\[
\begin{array}{l}
P_{1}=\{(a, 1),(b, 3),(c, 1),(c, 4),(c, 3),(c, 2)\} ; \\
P_{2}=\{(1,1),(1,2),(1,4),(2,1),(2,2),(2,3),(3,3), \\
(3,2),(3,4),(4,3),(4,4),(4,1)\} .
\end{array}
\]