Let R be the relation defined on Z by aRb if a + b is even. I have the following equivalence classes but have no clue how and why, so please really explain step by step how they get there in detail, Now note that the sum of two even numbers is even (2k +21 = 2(k+l))and the sum of two odd numbers is even ((2k+1)+(21+1) = 2(k+l+1))but the sum of an odd number with an even number is odd ((2k+1)+2l =2(k+1)+1). Therefore the equivalence classes are the set of odd numbersand the set of even numbers:{2k | k = Z}{2k + 1 | k ¤ Z}

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Let R be the relation defined on Z by aRb if a + b is even. I have the following equivalence classes but have no clue how and why, so please really explain step by step how they get there in detail,

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let R be the relation defined on Z by aRb if a + b is even. I have the following equivalence classes but have no clue how and why, so please really explain step by step how they get there in detail,

Now note that the sum of two even numbers is even (2k +21 = 2(k+l))
and the sum of two odd numbers is even ((2k+1)+(21+1) = 2(k+l+1))
but the sum of an odd number with an even number is odd ((2k+1)+2l =
2(k+1)+1). Therefore the equivalence classes are the set of odd numbers
and the set of even numbers:
{2k | k = Z}
{2k + 1 | k ¤ Z}

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