Equivalence Relations

An equivalence relation must be reflexive, symmetric, and transitive, a definition central to partitioning sets in JEE Advanced. Reflexivity: a-a = 0 is divisible by 5, so aRa holds for every integer. Symmetry: if a-b is divisible by 5 then b-a = -(a-b) is also divisible by 5, so bRa holds. Transitivity: if a-b and b-c are each multiples of 5, their sum (a-b)+(b-c)=a-c is a multiple of 5, so aRc holds. All three properties are satisfied, making R the congruence-modulo-5 relation that partitions integers into five residue classes. Option reflexivity-fails is false because zero is divisible by 5. Option symmetry-fails is false since divisibility is preserved under negation. Option transitivity-fails is false because sums of multiples of 5 remain multiples of 5. Hence R is a genuine equivalence relation. Plausibility check: the five classes {...,-5,0,5,...} through {...,-1,4,9,...} are disjoint and cover all integers, exactly as an equivalence relation requires. Equivalence relations and partitions are two faces of the same structure, since every partition of a set induces an equivalence relation and every equivalence relation produces a partition into disjoint classes. Treating congruence modulo a fixed integer as the canonical prototype lets a student verify reflexivity, symmetry and transitivity almost mechanically, which is exactly the efficiency JEE Advanced rewards under time pressure.