Question

If V={(x,y)∈R^2 | x=y} , then (1,0) +V and (0,1)+V are two distinct element of R^2/V.

True or false with full explanation

Answer 1

If (1,0) +V contained (0,1) then (0,1)-(1,0)=(-1,1) would belong to V.

But this is not the case, because of -1 is not equal to 1.

So, (1,0) +V contained (0,1) does not contain (0,1) and therefore, the classes (1,0) +V and (0,1) +V are distinct as elements of R²/V.