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.
