Define a relation R on Z by a R b if a = b + 5k for some integer k.Complete the following proof that R is reflexive.Let a ∈ ______. Since a = a + 5·______, a R a, as required.Complete the following proof that R is symmetric.Let a, b ∈ Z and suppose that a R b. By the definition of R, a = _____ for some integer k. Using algebra, b = ________, which shows that ______, as required.Complete the following proof that R is transitive.Let a, b, c ∈ Z and suppose that ______ and ______. By the definition of R, a = ______ for some integer k1 and b = ______ for some integer k2. Substituting the second equation into the first gives a = _________________ = c + 5(_____), which shows that _______, as required.

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Answered: Define a relation R on Z by a R b if a…

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Define a relation R on Z by a R b if a = b + 5k for some integer k.

Complete the following proof that R is reflexive.
Let a ∈ ______. Since a = a + 5·______, a R a, as required.
Complete the following proof that R is symmetric.
Let a, b ∈ Z and suppose that a R b. By the definition of R, a = _____ for some integer k. Using algebra, b = ________, which shows that ______, as required.
Complete the following proof that R is transitive.
Let a, b, c ∈ Z and suppose that ______ and ______. By the definition of R, a = ______ for some integer k1 and b = ______ for some integer k2. Substituting the second equation into the first gives a = _________________ = c + 5(_____), which shows that _______, as required.

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