Reducing $ab' + cb + ac$ to $ab' + cb$

Question

Boolean expressions $I = ab' + cb + ac$ and $J = ab' + cb$ have the same truth table. Then why expression $I$ can't be reduced to expression $J$?

Answer 1

Why do you think $I$ can't be rewritten to $J$? Here's one way to do it algebraically:

$$\begin{align} I &= ab'+cb+ac \\&= ab'+cb+ac(b+b') \\&= ab'+cb+acb+acb' \\&= ab'+acb'+cb+acb \\&= (a+ac)b'+(c+ac)b \\&= ab' + cb &= J \end{align}$$

Answer 2

Depending on the value of $b$, $J$ is either $a$ or $c$. But $a$ and $c$ must be true if $ac$ is true. Therefore, $ac$ does not add anything to expression $I$.

A more detailed explanation and historic background can be found in this article on the Consensus Theorem. Also have a look at a related post.