Boolean algebra how simplify products of sum Form

Question

How Solve it to minimum number of literals
i can't understand basic properties to simplify this expression
$(A̅ +C)(A̅ +C̅ )(C+D)(B̅ +D)(A+B+C̅ D)(A+B̅ +C)$
explain me to understand concepts of simplification!

Answer 1

Do it in parts.

$$ (\overline A + C)(\overline A + \overline C) $$

So multiply it out. 8b Distributive Law: $(W + X) • (Y + Z) = W Y + W Z + X Y + X Z$

$$\overline A\ \overline A + \overline A\ \overline C + \overline A C + \overline C C $$ 4a Complement Law: $ X • \overline X = 0 $

3a Idempotent Law: $ X • X = X $

$$\overline A + \overline A\ \overline C + \overline A C + 0 $$

2b Identity Law: $ X + 0 = X $

Extract common terms. 8a Distributive Law: $X • (Y + Z) = X Y + X Z$

$$\overline A ( 1 + \overline C + C) $$ $$\overline A $$

1b Annulment Law: $ X + 1 = 1 $

Expression becomes:

$$\overline A (C+D) (\overline B + D) (A + B + \overline C D) (A + \overline B + C)$$

Simplify the next two terms. Then the last two terms. Finally simplify the remainders.

Admit-ably, I did the easier part. Get familiar with a set of rules your instructor gave you. Then practice.

Laws and Theorems of Boolean Algebra