6.2.25 Binary relations

Condition: For binary relations \( \rho \) and \( \tau \) given on the set \( A=\{1,2,3,4,5\} \): a) write down matrices and construct graphs: b) use the matrix method to find the composition \( \rho \circ \tau \), construct a graph of the relation; d) explore the properties of the relations \( \rho, \tau \) and \( \rho \circ \tau \) (reflexivity, irreflexivity, symmetry, antisymmetry, transitivity). \[ \begin{array}{l} \rho=\{(x, y): x+y=0(\bmod 2)\} \\ \tau=\{(x, y): 2 \leq y \leq x-1\} \end{array} \]