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.

Answer 1

Answer with Step-by-step explanation:

We are given that an equivalence relation P on Z as

Let $x,y\in Z$

$xPy$ if and only if $k\in Z$ 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

$x-x=0=2(0)$

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

$x-y=2k$ where k is any integer

$y-x=-2k=2(-k)$

k belongs to integers.

Hence, relation satisfied  symmetric property.