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Consider the relation R on Z such that mRn if and only if 3 divides the difference m - n for m, n in Z.

Remark: Recall, in general, for l, k in Z, we say l DIVIDES k if k = l * q, that is, r = 0 in the division algorithm or has remainder 0 after division.

1. Prove that R is reflexive.
2. Prove that R is symmetric.
3. Prove that R is transitive.
4. Describe the equivalence classes of R.
5. What is the quotient set with respect to R? Finding convenient symbols for its equivalence classes and writing the set of these will suffice to answer the question.

Sagot :

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