[Math] Boolean Simplification Methods. : Karnaugh Maps vs Boolean Algebra

Question

I am practicing problems on simplifying Boolean Expressions with Boolean Algebra and Karnaugh Maps.

• I tried to simplify the same Boolean Expression using both ways
  but I got two different answers. Can that happen?
• After any Boolean Expression simplification, whatever may be the method, the answers should be same, right? Or can the answers be different?

I tried simplifying the expression
$$\mathcal{A'B'C'+A'B'C+A'BC'+ABC'+ABC}$$
With boolean algebra I got $\mathcal{A'B'C+AB}$, and with Karnaugh Maps I got $\mathcal{A'+B}$

Answer 1

Both Karnaugh Map and Boolean Algebra Simplification need not to give same answer. The answer may differ.

The Boolean Algebra Simplification is sometimes tricky because we need smart use of Properties (Absorption and Distributive) and Theorems (Redundancy Theorem)

Even K-Map Solutions are not unique. The answer may differ based on your choice of pairing. The same set of minterms can be simplified in two ways as shown in this attached image.

Now, considering your problem, all the three expressions are not equivalent.

Let three expressions be
$\mathcal{F=A′B′C′+A′B′C+A′BC′+ABC′+ABC}$
$\mathcal{K=A′B′C+AB}$
$\mathcal{S=A′+B}$

$\mathcal{A}$	$\mathcal{B}$	$\mathcal{C}$	$\mathcal{F}$	$\mathcal{K}$	$\mathcal{S}$
0	0	0	1	0	1
0	0	1	1	1	1
0	1	0	1	0	1
0	1	1	0	0	1
1	0	0	0	0	0
1	0	1	0	0	0
1	1	0	1	1	1
1	1	1	1	1	1

Clearly, $\mathcal{F}$, $\mathcal{K}$ and $\mathcal{S}$ aren't equivalent.

Thus, the claimed simplified expression (using K-Map and Boolean) are actually incorrect.

The actual simplified expressions are :

$\mathcal{AB+A′B′+A′C′}$

$\mathcal{AB+A′B′+BC′}$