Question

Define a binary relation S on the set ? of all real numbers by declaring that for all a,b ? ?, aSb ? ?k ? ? : a = b + k. That is, aSb if and only if a = b + k for some integer k. Show that S is an equivalence relation on the set ?. (Suggestion: Use methods similar to those used in the proof that the relation of divisibility, |, on ? is reflexive and transitive; this question is very similar to the Homework Exercise on proving that ?? is an equivalence relation.)

          Define a binary relation S on the set ? of all real numbers by declaring that for all a,b ? ?,
aSb ? ?k ? ? : a = b + k.
That is, aSb if and only if a = b + k for some integer k.
Show that S is an equivalence relation on the set ?.
(Suggestion: Use methods similar to those used in the proof that the relation of divisibility, |, on ? is reflexive and transitive; this question is very similar to the Homework Exercise on proving that ?? is an equivalence relation.)

Define a binary relation S on the set ? of all real numbers by declaring that for all a,b ? ?,
aSb ? ?k ? ? : a = b + k.
That is, aSb if and only if a = b + k for some integer k.
Show that S is an equivalence relation on the set ?.
(Suggestion: Use methods similar to those used in the proof that the relation of divisibility, |, on ? is reflexive and transitive; this question is very similar to the Homework Exercise on proving that ?? is an equivalence relation.)

Added by Erik S.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

06/23/2023

Step 1

Reflexivity: We need to show that for all a ∈ R, aSa. This means we need to find an integer k such that a = a + k. We can choose k = 0, since a + 0 = a. Therefore, aSa for all a ∈ R, and S is reflexive. Show more…

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Define a binary relation S on the set ℐ of all real numbers by declaring that for all a,b ∈ ℐ, aSb ℔ ∃k ∈ ℤ : a = b + k. That is, aSb if and only if a = b + k for some integer k. Show that S is an equivalence relation on the set ℐ. (Suggestion: Use methods similar to those used in the proof that the relation of divisibility, |, on ℤ is reflexive and transitive; this question is very similar to the Homework Exercise on proving that ≡ₙ is an equivalence relation.)