Let F be a field with 4 elements {0,1,x,y}. Here 0,1 are the additive and multiplicative neutral element, as usual.

a) Prove xy=1.

b) Prove xx=y and yy=x.

c) Prove x+y=1.

d) Prove 1+1=0, x+x=0, y+y=0, x+1=y, and y+1=x.

(You can also do parts (a)-(d) in a different order, if you wish.)