Let $R$ be the relation on $Z($the set of integers$)$ defined by $(x,y)inR$ iff $x^2+y^2=2k$ for some integers $k0.R$ is symmetric. Which one of the following is a valid proof showing that $R$ is symmetric?

a$.$ Let $x,$yinZ$$ be given.

Suppose $(x,y)inR$ then $x^2+y^2=2k$ for some $k0.$ ie $y^2+x^2=2k$ for some $k0.$ thus $(x,y)inR.$

b$.$ Let $x,$yinZ$$ be given. Suppose $(x,y)inR$ then $x^2+y^2=2k$ for some $k0.$ ie $y^2+x^2=2k$ for some $k0.$ thus $(y,x)inR.$

c$.$ Let $x,$yinZ$$ be given.

Suppose $(x,y)inR$ then $x^2+y^2=2k$ for some $k0.$ thus $(y,x)inR.$

d$.$ Let $x,$yinZ$$ be given.

Suppose $(x,x)inR$ then $x^2+y^2=2k$ for some $k0.$ ie $y^2+x^2=2k$ for some $k0.$ thus $(y,y)inR.$