Problem:
Solution:
Instead of exhausting the formula, it is easier to describe and prove by defining those operation as mod 2 operations.
Now addition mod 2 is closed, associative, commutative, has identity 0 and inverse well defined.
Multiplication is closed, associative, commutative and distributive under addition.
The only non trivial thing is multiplicative inverse, but that is trivial for mod 2 because the only element need a multiplicative is 1, which is simply itself.
Solution:
Instead of exhausting the formula, it is easier to describe and prove by defining those operation as mod 2 operations.
Now addition mod 2 is closed, associative, commutative, has identity 0 and inverse well defined.
Multiplication is closed, associative, commutative and distributive under addition.
The only non trivial thing is multiplicative inverse, but that is trivial for mod 2 because the only element need a multiplicative is 1, which is simply itself.
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