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Unit 3: Boolean Algebra

truth table specification! The steps to take from a truth table to the final circuit are so unambiguous Notes
and direct that it requires little, if any, creativity or other original thought to execute them.

Be careful that “Truth table” must be considered while designing the digital
circuits.

Stone’s representation theorem for Boolean algebra states that every Boolean
algebra is isomorphic to a field of sets, as stated by Marshall Harvey Stone
in 1936.
3.1.5 Boolean Rules for Simplification

Boolean algebra finds its most practical use in the simplification of logic circuits. If we apply
certain algebraic rules to a Boolean equation resulting from a truth table, we will get a simpler
equation. The simplified equation may be translated into circuit form for a logic circuit performing
the same function with fewer components. If equivalent function may be achieved with fewer
components, the result will be increased reliability and decreased cost of manufacture. A few of
the Boolean rules for simplification are shown in Figure 3.6.

Figure 3.6: Useful Boolean Rules for Simplification
A + AB = A
A + AB = A + B

(A + B) (A + C) = A + BC
Proving the rules above requires the use of concepts learned in sections 5 (a), (b) and (c). It proves
the first two below and leave the third as an exercise.

A + AB = A.(1 + B) [Distributive property]
= A
A + A’B = (A + AB) + A’B [From rule above: A = A+B]
= A + (AB + A’B) [Additive association property]

= A + B.(A + A’) [Distributive property]
= A + B
Now, we will encapsulate what we learned about the binary number systems, logic gates and
Boolean algebra by designing a few common building blocks of digital systems.

3.2 Boolean Theorems

We have seen how Boolean algebra can be used to help analyze a logic circuit and express its
operation mathematically. We will continue our study of Boolean algebra by investigating the
various Boolean theorems (rules) that can help us to simplify logic expressions and logic circuits.
The first group of theorems is given in Figure 3.7. In each theorem, x is a logic variable that can
be either a 0 or a 1. Each theorem is accompanied by a logic-circuit diagram that demonstrates
its validity.
Theorem (1) states that if any variable is ANDed with 0, the result has to be 0. This is easy to
remember because the AND operation is just like ordinary multiplication, where we know that
anything multiplied by 0 is 0. We also know that the output of an AND gate will be 0 whenever
any input is 0, regardless of the level on the other input.

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