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+1 vote
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in Sets, Relations and Functions by (44.0k points)
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Show that the following are equivalence relation:

R in A = {x ∈ Z | 0 ≤ x ≤ 12} given by R = {(a, b) / |a – b| is a multiple of 4}

1 Answer

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by (44.1k points)
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Best answer

a. Since |a – a| is a multiple of 4,

(a, a) ∈ R

∴ R is reflexive.

b. Let (a, b) ∈ R

Then a – b = ±4m,

∴ b – a = ±4m, where m is an integer

∴ (b, a) ∈ R

∴ R is symmetric.

c. Let (a, b), (b, c) ∈ R

a – b = ± 4m, b – c = ± 4n,

∴ a – c = ±4(m + n), where m, n are integers

∴ (a, c) ∈ R

∴ R is transitive

Thus, R is an equivalence relation.

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