For each of the following relations, either prove that it is an equivalence rela

Question

For each of the following relations, either prove that it is an equivalence relation or prove that it is not an equivalence relation.

(a) For integers a and b, a b if and only if a + b is even.

(b) For integers a and b, a b if and only if a + b is odd. Sec. 2.9 Exercises 47

(c) For nonzero rational numbers a and b, a b if and only if a × b > 0.

(d) For nonzero rational numbers a and b, a b if and only if a/b is an integer.

(e) For rational numbers a and b, a b if and only if a b is an integer.

(f) For rational numbers a and b, a b if and only if |a b| 2.

Explanation / Answer

Answer:

a) This is an equivalence that effectively splits the integers into odd and even sets. It is reflexive (x + x is even for any integer x), symmetric (since x + y = y + x) and transitive (since you are always adding two odd or even numbers for any satisfactory a, b, and c).
(b) This is not an equivalence. To begin with, it is not reflexive for any integer.
(c) This is an equivalence that divides the non-zero rational numbers into positive and negative. It is reflexive since xx > 0. It is symmetric since xy = yx. It is transitive since any two members of the given class satisfy the relationship.

(d) This is not an equivalance relation since it is not symmetric. For exam- ple, a = 1 and b = 2.
(e) This is an eqivalance relation that divides the rationals based on their fractional values. It is reflexive since for all a, aa = 0. It is symmetric since if ab = x then ba = x. It is transitive since any two rationals with the same fractional value will yeild an integer.
(f) This is not an equivalance relation since it is not transitive. For exam- ple, 4 2=2 and 2 0=2, but 4 0=4.

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