What is wrong with the way this boolean expression is simplified?

Question

I've been trying to simplify this expression, and I am 99 percent sure the way i did it is wrong but I am not sure where I went wrong. It seems to be algebraically accurate to me, any help would be greatly appreciated.

Answer 1

You did everything corrctly, except in the last step you should get:

$\bar{B_1}(\bar{A_1}\bar{\color{red}{B_0}}+A_1)$

But this statement can be simplified a bit more:

In general:

$pq+\bar{p}=$ Distribution

$(p + \bar{p})(q+\bar{p})=$ Complement

$1(q+\bar{p})=$ Identity

$q+\bar{p}$

Indeed, do yourself a favor and remember these two equivalences that come up a lot when you do this kind of Boolean algebra:

Reduction

$pq+\bar{p}=q+\bar{p}$

$(p +q)\bar{p}=q\bar{p}$

The second of these is the most intuitive: you already know that $\bar{p}$ has to be true, but you also want $p+q$ to be true, i.e.one of $p$ and $q$ should be true. But since $p$ has to be false, that means $q$ has to be true. In other words, in the context of $\bar{p}$, the term $p+q$ reduces to $q$. The first equivalence is just the dual of the second, and is what I showed in a few steps before, but if you are in a hurry, you can do it now in 1 step using Reduction: a good rule to have!

So, le's apply Reduction to what we got in your case:

$\bar{B_1}(\bar{A_1}\bar{B_0}+A_1)=$ Reduction

$\bar{B_1}(\bar{B_0}+A_1)$

I bet that that is the answer you were looking for!