# Proving or disproving a basic equation in Boolean algebra

I need to prove or disprove that in any Boolean algebra: if $$a+ab=b$$ then $$a=b=1$$ or $$a=b=0$$.

I build the following truth table: $$\begin{array}{|c|c|c|} \hline a & b & a+ab \\ \hline 0 & 0 & 0 \\ \hline 0 & 1 & 0 \\ \hline 1 & 0 & 1 \\ \hline 1 & 1 & 1 \\ \hline \end{array}$$ So it does looks like that theorem is true. Can I prove it with algebra? if not, How should I prove it?

Edit: You guys proved it for the binary Boolean algebra. The theorem is for every Boolean algebra (I just gave an example for binary). How can I prove it for every Boolean algebra?

• Yes, the statement holds in the two-element Boolean algebra. But that's not what the exercise is asking. Oct 1 '18 at 13:59
• Look into any Boolean algebra with more than two elements (for example the four-element Boolean algebra) and check what happens if $a=b$ but they are neither $0$ nor $1$. Oct 1 '18 at 13:59

$a + ab = b \iff a(1+b) = b \iff a = b$
Adding $$b+1$$ to both sides of $$a+ab=b$$, you get $$\require{cancel} ab+a+b+1=\underbrace{\cancel{b+b}}_{=2b=0}+1$$ (recall a Boolean ring has characteristic 2). Thus $$(1+a)(1+b)=1$$. This only holds if $$1+a=1+b=1$$, i.e., $$a=b=0$$.
• Looks like you're using the wrong interpretation of addition, which would make the question wrong too. In the question, as in classical logic, $+$ denotes disjunction, not XOR. Oct 1 '18 at 15:15
• Sensible people always write $\vee$ when they mean disjunction or lattice join. If $+$ means disjunction (which is illogical) then there is no point in performing the "+ab", and any a=b in any Boolean algebra works, not just =0 or =1. Oct 1 '18 at 15:24
• Well, many introductory texts use "$+$" for disjunction in classical boolean algebra. Whether we like it or not, it is the norm at lower levels, not "$\lor$". As for "no point", many exercises have no point. Anyway, it's up to the asker to decide whether your answer is useful or not. I was just pointing out that you're not interpreting "$+$" in the question in the intended manner (as is clear from the truth-table). If you take offense with my use of "wrong" in my comment, please understand that I did not mean that your interpretation is mathematically wrong or inferior, just different. Oct 1 '18 at 15:50