For A={0,1,2,3,4,5} and R={(a,b): a²≡b² (mod 6)}, how many equivalence classes does R have?

Correct answer: 4

Direct answer: C, four equivalence classes. Compute the square residues modulo 6: 0² gives 0, 1² gives 1, 2² gives 4, 3² gives 3, 4² gives 16≡4, and 5² gives 25≡1. Thus the elements with the same square residue group together: residue 0 gives {0}; residue 1 gives {1,5}; residue 4 gives {2,4}; and residue 3 gives {3}. These four disjoint groups cover A, so the equivalence classes are {0}, {1,5}, {2,4}, and {3}. Option A, 2, misses two groups; B, 3, also misses one; D, 6, incorrectly treats every element as separate. The relation is an equivalence relation because equality of the values a² modulo 6 is reflexive, symmetric, and transitive. Memory cue: first calculate the invariant—in this case the square residue—then group equal values.