Prove that $\neg 0 = 1$ in every boolean algebra.

Question

I am trying to prove that $\neg 0 = 1$ in every boolean algebra by using the laws of boolean algebra.

The assistant told us to start by using the identity law and form $\neg 0 = 1$ to $\neg 0 = \neg 0 \cdot 1$. However I don't know how to proceed from there.

Answer 1

Identity Law: $\lnot 0 = \lnot0\circ 1$

Double Negation: $\lnot 0 \circ 1 = \lnot 0 \circ \lnot \lnot 1$

De Morgan: $\lnot 0 \circ\lnot \lnot 1 = \lnot(0 + \lnot 1)$

Indentity: $\lnot (0 + \lnot 1) = \lnot \lnot 1$

Double Negation: $\lnot\lnot 1 = 1$.

....

Oh this is better: $\lnot 0 + 0 = 1$ by the complement law.

But $\lnot 0 + 0 = \lnot 0$ by the identity law.

Answer 2

Well, think about the defining property of $\neg$ in a Boolean algebra: $$(\neg x)\vee x=1.$$ Taking $x=0$, this gives $(\neg 0)\vee 0=1$.

Now, you know a rule for simplifying expressions of the form $a\vee 0$. Can you see how to use it here to get what you want?