How to Convert in Boolean Algebra

Boolean logic may be expressed in several different terms. Boolean algebra is one of those ways. You can use Boolean algebra to simplify complicated logic by applying some simple theorems and rules. Understanding how to convert Boolean expressions from one form to another and how to simplify Boolean expressions to their simplest terms using algebra is an important skill for anyone who is interested in logic-based programming or designing electrical circuitry.

Instructions

Converting a Boolean Function From Truth Table to SOP Algebraic Equation

1. 

   Create a truth table from your circuit diagram or logical assumptions as shown in Illustration 1. In this example, we're using a Boolean expression with three variables; A, B and C.

2. Write a Boolean product expression for each row where the output column equals 1.
   In simple terms, the output is true (or high) if any of the following statements are true:
   B and C are true, A is NOT true..
   A and B are true, C is NOT true.
   A and C are true. B is NOT true.
   Using the chart in the example above, the Boolean expressions would be:
   A'BC ABC' AB'C ABC
   A, B and C are true.

3. Join the product expressions by adding them with a + operator. Continuing the example, the Boolean equation would be
   OUTPUT = A'BC + ABC' + AB'C + ABC

Simplify Boolean Expressions Using Algebra

1. Evaluate your expression to decide the best way to proceed. Simplifying Boolean expressions requires a familiarity with algebraic properties and Boolean theorems. The following steps demonstrate how to simplify a specific Boolean equation with an explanation of each step.

2. Factor out like terms to simplify the original Boolean expression. Example:
   A'BC + AB'C + ABC' + ABC = OUTPUT
   Because BC appears in both A'BC and ABC, you can combine the two statements to make BC(A + A')
   BC(A + A') + AB'C + ABC' = OUTPUT

3. Apply Boolean identity theorem to the simplified Boolean expression. Example:
   BC(A + A') + AB'C + ABC' = OUTPUT
   Because A + A' (A OR NOT A) is always 1, you can apply the identity theorem to make:
   BC(1) + AB'C + ABC' = OUTPUT
   Because A1=A, you can apply the identity theorem to convert the above equation to:
   BC + AB'C +ABC' = OUTPUT

4. Factor out the next variable. Example:
   BC + ABC' + AB'C = OUTPUT
   Factor out the B in the first two expressions to simplify by using the associative property like so: BC + BAC'=B(C +AC') which converts the above equation to:
   B(C + AC') + AB'C = OUTPUT

5. Apply A + A'B = A + B theorem where appropriate. Example:
   B(C + AC') + AB'C = OUTPUT
   C + AC' is the same as C + C'A. C + C'A is the same as C + A. Using that theorem, you can convert the above statement to:
   B(C + A) + AB'C = OUTPUT -

6. Use the distributive property to simplify the Boolean expression. Example:
   B(C + A) + AB'C = OUTPUT
   B(C + A) is the same as BC + BA, thus:
   BC + BA + AB'C = OUTPUT

7. Factor out the A variable by combining terms in which it appears. Example:
   BC + BA + AB'C = OUTPUT
   BC + A(B + B'C) = OUTPUT

8. Repeat Step 5 to simplify the second term in the Boolean equation. Example:
   BC + A(B + B'C) = OUTPUT
   B + B'C = B + C, thus you can convert the above statement to:
   BC + A(B + C) = OUTPUT

9. Use the distributive property to rewrite the Boolean equation in its simplest terms. Example:
   BC + A(B + C) = OUTPUT
   becomes:
   BC + AB + AC = OUTPUT
   or
   AB + BC + AC = OUTPUT
   In simple language, the Boolean equation states that if A and B are true OR B and C are true OR A and C are true, then the statement evaluates to TRUE or 1.

Tips & Warnings

• Use the FOIL (First, outer, inner, last) principle to help you decide on the order of operations. Factor variables out of your equation wherever possible. Continue simplifying until you reach the simplest possible Boolean expression for your purposes.

• Remember that the + operator in Boolean algebra means OR and the * operator means AND.

Resources

• Learn more about Boolean Algebra at Web-Books.com