Logic and sets · Relations · Equivalence relations · Partitions
On the set
\[ A=\left\{0,\;1,\;-1,\;2,\;\tfrac12,\;-2,\;-\tfrac12,\;3,\;\tfrac13\right\} \]
define the relation
\[ \rho=\left\{(a,b)\in A\times A \;:\; \left(a^{2}-b^{2}\right)(ab-1)=0\right\}. \]
(a) Decide which of the properties reflexive, symmetric, antisymmetric, transitive the relation \(\rho\) has.
(b) Decide whether \(\rho\) is an equivalence relation. If it is not, exhibit at least one set \(\rho_{1}\subset A\times A\) for which \(\rho\cup\rho_{1}\) is an equivalence relation.
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Serbian Regional Competition (Okruzno takmicenje) 2026, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source