Question

Determine the distance equivalence classes for the relation R is defined on ℤ by a R b if |a - 2| = |b - 2|.

I had to prove it was an equivalence relation as well, but that part was not hard. Just want to know if the logic and presentation is sound for the last part:

8.48) A relation R is defined on ℤ by a R b if |a - 2| = |b - 2|. Prove that R is an equivalence relation and determine the distinct equivalence classes.

Proof We show that R is reflexive, symmetric, and transitive.

Reflexivity: Let a ∈ ℤ. a R a since it is the case that |a - 2| = |a - 2| for all a.

Symmetry: Suppose a R b. Then |a - 2| = |b - 2|. |b - 2| = |a - 2| trivially.

Transitivity: Suppose a R b and b R c. |a - 2| = |b - 2| and |b - 2| = |c - 2|. a R c by transitive property of equality.

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Some equivalence classes are:

...[-3] = {-3, 7}, [-2] = {-2, 6}, [-1] = {-1, 5}, [0] = {0, 4}, [1] = {1, 3}, [2] = {0}, [3] = {1, 3}, [4] = {0, 4}, [5] = {5, -1}, [6] = {6, -2}, [7] = {7, -3}, [8] = {8, -4}, [9] = {9, -7}...

Observe that the distinct equivalence classes are [0], [1], [2], [4], [5], [6], [7], [8], [9], ..., [n]

Define the set of distinct equivalence classes as N = {[n] : n ∈ Z+ - {3}}

Answer 1