Boolean bit OR operation on a Finite Field

Question

How can I express $x \vee y$ in $GF_2$?

I know that XOR is $GF_2[x]+GF_2[y]$ and AND is $GF_2[x]*GF_2[y]$ for instance.

But I cannot figure out bitwise disjunction.

This may be because OR does not have an inverse element and is therefore not a finite field.

Still, does it have a simple expression in $GF_2$.

Answer 1

One way you could express it is by $$x\vee y = xy + x + y.$$

Answer 2

$$x\lor y=\lnot((\lnot x)\land(\lnot y))=1-((1-x)\cdot(1-y))=x+y-xy=x+y+xy.$$