Define an equivalence relation P on Z as follows: Let x,y e Z ; xPy if and only if ke Z s.t. x - y = 2k Show how the reflexive property and symmetric property of equivalence relations hold for P on Z.

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Question:

Define an equivalence relation P on Z as follows: Let x,y e Z ; xPy if and only if ke Z s.t. x - y = 2k Show how the reflexive property and symmetric property of equivalence relations hold for P on Z.

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Answer with Step-by-step explanation:We are given that an equivalence relation P on Z as Let [tex]x,y\in Z[/tex][tex]xPy[/tex] if and only if [tex]k\in Z[/tex] such that x-y=2k.We have to show that how the reflexive property and symmetric property of an equivalence relations hold for P on Z.We know that reflexive property a is related to a by given relations.If xPax then we get [tex]x-x=0=2(0)[/tex]Where k=0 and 0 belongs to integers.Hence, the relation satisfied reflexive property.Symmetric property :If a is related to b then b is related to b.If x and y is related by the relation [tex]x-y=2k[/tex] where k is any integer[tex]y-x=-2k=2(-k)[/tex]k belongs to integers.Hence, relation satisfied  symmetric property.

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