BOOLEAN ALGEBRA FOR BEGINNERS.

Boolean Algebra

A variable used in an algebraic formula so far, is assumed to

take a set of numerical values. All variables in Boolean equations can take only one of two

possible values. Used symbols for the two values are 0 and 1. Rules first defined for logic by George Boole (1854), were adapted for the use in designing electronic circuits. The circuits in computers and other electronic devices have inputs, each of which is either a 0 or a 1. One major advantage in using these rules is to simplify an electronic circuit. Boolean algebra provides the operations and the rules for working with Boolean variables.

               Three (3) Boolean operators are discussed.

                   • Complement

                   • Boolean sum

                   • Boolean product

Ten (10) rules are also discussed (aka Boolean Identities).

06/02/2017  

Boolean Operators

  Complement

  Defined as the opposite of the value that a Boolean variable takes. Denoted with a bar (E.g.:Ā).

  Boolean Sum

  Defined as the output to be 1 if at least one variable is 1. Denoted with the symbol + or by OR.

  0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1 and 1 + 1 = 1.

  Boolean Product

  Defined as the output to be 0 if at least one variable is 0. Denoted with the symbol ( ∙ ) or by AND.

  0 ∙ 0 = 0, 0 ∙ 1 = 0, 1 ∙ 0 = 0 and 1 ∙ 1 = 1.

  When there is no danger of confusion, the symbol ∙ can be omitted.

          Order of Boolean operators,

                1. Complement.

                2. Boolean products.

                3. Boolean sums.

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Truth Tables

  

To verify the above rules, a truth table can be used. It’s also known as a Table of Combinations. It’s a table displaying all possible values for the variables and the outcomes for a Boolean expression. If there are n number of variables, there will be n power of 2 number of rows in the truth table. If the truth table for two Boolean expressions shows the same outcomes for the same values for the variables, it can be concluded that the expressions are the same/equal.

Sum of Products (SOP)

In some cases, the truth table might be known and we might want to know the expression that gives the truth table. This can be done by representing as a Sum of Products (SOP) of the variables and their complements.

  Steps:-

1. Select the rows in the truth table that gives 1 as the outcome.

2. Write how we can obtain 1 for the first selected row by using the

product of the variables.

3. Repeat step two for all selected rows and use the sum to combine

all results.

Product of Sums (POS)

  Used for the same reason as a SOP. Product of Sums (POS) has opposite steps of SOP.

  Steps:-

1. Select the rows in the truth table that gives 0 as the outcome.

2. Write how we can obtain 0 for the first selected row by using the

sum of the variables.

3. Repeat step two for all selected rows and use the product to

combine all results.

Conversion can be done between the two using De Morgan’s rule.

Duality Principle

  

In a Boolean expression, if all the sums (+) and products ( ∙ ) are exchanged as well as if 1’s and 0’s are exchanged, the resulting expression is the opposite of the initial expression. This property is observed between SOP and POS. The duel of the complement of one form is equal to the expression in the other form.

Summary 

 Should be able to,

• Understand the Boolean expressions.

• Learn laws and rules of Boolean algebra.

• Simplify Boolean expressions using Boolean identities.

• Use Sum of Products (SOP) and Product of Sums (POS) to find Boolean expressions.

• Understand similarities and differences between Boolean variables as opposed to regular variables.

Next Lecture:- Logic Gates