Write the boolean expression from OR-AND to AND-OR

Question

I have an expression (a+b'+c')(a'+b'+c) and I am struggling to rewrite it as an AND-OR from its current OR-AND state. I started using De Morgans, but that went the wrong way quickly.

How can I make this an AND-OR using properties and theorems of Boolean algebra?

Answer 1

Just use distribution:

$$(a+b'+c')(a'+b'+c)=$$

$$aa'+ab'+ac+b'a'+b'b'+b'c+c'a'+c'b'+c'c=$$

$$0+ab'+ac+b'a'+b'+b'c+a'c'+b'c'+0=$$

$$ac+b'+a'c'$$

You could also have done:

$$(a+b'+c')(a'+b'+c)=$$

$$b'+(a+c')(a'+c)=$$

$$b'+aa'+ac+c'a'+c'c=$$

$$b'+ac+a'c'$$

Answer 2

We begin with a conjunction of disjunctions (conjunctive normal form, CNF) $$(a+b'+c')(a'+b'+c)$$

We should first notice the common disjunct $b'$ $$(a+c')(a'+c)+b'$$

Distributing the rest will then give you a disjunction of conjunctions (disjunctive normal form, DNF) as required.   Vanishing the conjoined complements ($aa', cc'$) will simplify this to a disjunction of three conjuncts.