Logic and sets · Relations · Equivalence relations · Order relations · Equivalence classes
Consider the set of five Serbian words (written in the Latin alphabet)
and define two relations on \(X\): for words \(x, y \in X\),
\(x \mathrel{\rho_1} y\) if and only if \(x\) and \(y\) have the same number of letters;
\(x \mathrel{\rho_2} y\) if and only if \(x\) and \(y\) end in the same letter.
(a) For each of \(\rho_1\) and \(\rho_2\), determine whether it is reflexive, symmetric, antisymmetric, transitive.
(b) For each of \(\rho_1\) and \(\rho_2\), decide whether it is an equivalence relation and whether it is an order relation. For every equivalence relation among them, find all of its equivalence classes.
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Serbian Municipal Competition 2023, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source