Let R be the equivalence relation on the set Z of all integers given by \(R = \{(a, b) : a - b \text{ is divisible by } 7\}\). The number of distinct equivalence classes of R is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 7
Two integers a and b are related exactly when \(a - b\) is divisible by 7, that is, when a and b leave the same remainder on division by 7.
The possible remainders are
0, 1, 2, 3, 4, 5, 6
Each remainder gives one class, for example \([0] = \{\ldots, -7, 0, 7, 14, \ldots\}\) and \([1] = \{\ldots, -6, 1, 8, 15, \ldots\}\).
So there are 7 distinct equivalence classes: [0], [1], [2], [3], [4], [5], [6].