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Abstract Algebra : Equivalence Relations

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A relation R is defined on the set Z of all integers. In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and list at least four members of each.

1- xRy if and only if x^2+y^2 is a multiple of 2.
**Write x^2+y^2 as (x+y)^2-2xy

2- xRy if and only if x+3y is a multiple of 4 .

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Equivalence relations are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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A relation R is defined on the set Z of all integers. In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and list at least four members of each.

1- xRy if and only if x2 + y2 is a multiple of 2.
Reflexive: x2 + x2 = 2x2 is clearly a multiple of 2, so xRx
Symmetric: x2 + y2 is a multiple of 2 implies that y2 + x2 is a multiple of two.
So xRy  yRx
Transitive: x2 + y2 is a multiple of 2 and y2 + z2 is a multiple of 2
 x2 + y2 + y2 + z2 is a ...

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