I am trying to simplify (minimise) a Boolean expression but I can't seem to get the result I am supposed to. I even ran into a pretty bad contradiction and I have no idea what's wrong...
This is my expression:
(spacing between letters means multiplication (AND) and the $+$ operator means addition (OR))
$$ \mathrm{F}2 = \overline{A} \: \overline{B} \: \overline{C} + \overline{A} B \overline{C} + \overline{A} B C + A B \overline{C} $$
This is the way I simplified it:
$$ \begin{aligned} \mathrm{F}2 &= \overline{A} \: \overline{B} \: \overline{C} + \overline{A} B \overline{C} + \overline{A} B C + A B \overline{C} = \\ &= \overline{A} (\overline{B} \: \overline{C} + B \overline{C} + B C) + A B \overline{C} = \\ &= \overline{A} (\overline{C} (\overline{B} + B) + B C) + A B \overline{C}) = \\ &= \overline{A} (\overline{C} + B C) + A B \overline{C} = \\ &= \boxed{\overline{A} \: \overline{C} + \overline{A} B C + A B \overline{C}} \end{aligned} $$
However, if I enter the expression into a calculator, such as Logic Friday 1, I get this:
So basically it says the simplified expression should be
$$ \mathrm{F}2 = \boxed{\overline{A} B + \overline{A} \: \overline{C} + B \overline{C}} $$
But this would mean, looking at the two boxed (supposedly equal) expressions, that (since $\overline{A} \: \overline{C}$ is present in both)
$$ \overline{A} B C + A B \overline{C} = \overline{A} B + B \overline{C} $$
Which is quite impossible... Just take a look at this MS Office Excel table illustrating how they are different:
So what is going on here?
Could anybody please explain? :/
When we use logic operations we can't eliminate $\overline{A}\:\overline{C}$ just because they are present in both parts of the equation. $\overline{A}BC+AB\overline{C}+\overline{A}\:\overline{C}=1$ when $A=C=0$ and $B=1$. But $\overline{A}BC+AB\overline{C}=0$ for the same values of $A, B, C$