Boolean algebra finds its most practical use in
the simplification of logic circuits. If we translate a logic circuit's
function into symbolic (Boolean) form, and apply certain algebraic rules to
the resulting equation to reduce the number of terms and/or arithmetic
operations, the simplified equation may be translated back 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.
To this end, there are several rules of Boolean algebra presented in this
section for use in reducing expressions to their simplest forms. The
identities and properties already reviewed in this chapter are very useful
in Boolean simplification, and for the most part bear similarity to many
identities and properties of "normal" algebra. However, the rules shown in
this section are all unique to Boolean mathematics.
This rule may be proven symbolically by factoring an "A" out of the two
terms, then applying the rules of A + 1 = 1 and 1A = A to achieve the final
result:
Please note how the rule A + 1 = 1 was used to reduce the (B + 1) term to
1. When a rule like "A + 1 = 1" is expressed using the letter "A", it
doesn't mean it only applies to expressions containing "A". What the "A"
stands for in a rule like A + 1 = 1 is any Boolean variable or
collection of variables. This is perhaps the most difficult concept for new
students to master in Boolean simplification: applying standardized
identities, properties, and rules to expressions not in standard form.
For instance, the Boolean expression ABC + 1 also reduces to 1 by means
of the "A + 1 = 1" identity. In this case, we recognize that the "A" term in
the identity's standard form can represent the entire "ABC" term in the
original expression.
The next rule looks similar to the first on shown in this section, but is
actually quite different and requires a more clever proof:
Note how the last rule (A + AB = A) is used to "un-simplify" the first
"A" term in the expression, changing the "A" into an "A + AB". While this
may seem like a backward step, it certainly helped to reduce the expression
to something simpler! Sometimes in mathematics we must take "backward" steps
to achieve the most elegant solution. Knowing when to take such a step and
when not to is part of the art-form of algebra, just as a victory in a game
of chess almost always requires calculated sacrifices.
Another rule involves the simplification of a product-of-sums expression:
And, the corresponding proof:
To summarize, here are the three new rules of Boolean simplification
expounded in this section:
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