So I have this set theory expression that I'd like to show in boolean algebra. So if $A \cap B = A \cap C$, then I can show that $A \cap B^c = A \cap C^c$, since this is just $A \setminus B = A \setminus (A \cap B) = A \setminus (A \cap C) = A \cap C^c$.
But in terms of Boolean algebra, this would amount to showing that $ab = ac$ implies that $a\bar{b} = a\bar{c}$. I'm at a bit of a loss for how to do this, and would appreciate any thoughts.
Thanks.
(Just for my own curiosity).
You can start from the identity $\overline{xy} = \bar{x} + \bar{y}$. Now, since $ab = ac$, you have $\overline{ab} = \overline{ac}$ and thus \begin{align} a\bar{b} &= 0 + a\bar{b} = a\bar{a} + a\bar{b} = a(\bar{a} + \bar{b})\\ &= a(\overline{ab}) = a(\overline{ac})\\ & = a(\bar{a} + \bar{c}) = a\bar{a} + a\bar{c} = 0 + a\bar{c} \\ &= a\bar{c} \end{align}