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Boolean Algebra

All modern digital computers are based on binary system. It was developed by Chinese and Greek philosophers who created a methodology for binary codes known as propositional logics. It may be True or False and are the functions of three basic logical connectivity's and, or not.

**Various terms used in Boolean algebra are as follows-**

1. Variable- represents a logical quantity represented by 1 or 0 values.

2. Complement- it refers to the inverse of variable and denoted by a bar over variable.

**LAWS AND RULES OF BOOLEAN ALGEBRA**

1. Commutative law of addition for two variables is written as A+B = B+A.

This law states that the order in which the variables are makes no difference. Codes applied to logic circuits, addition and the OR operation are the similar.

2. The associative law of addition is written as follows for three variables: A + (B + C) = (A + B) + C This law states that when more than two variables are added the outcome is the same regardless the way or order in which we group the different codes.

3. The distributive law is written for three variables as A (B + C) = AB + AC.

This law states that two or more variables and the result with a single variable is equivalent to the single variable with each of the two or more variables and then products. While writing the code we can take out common factors for easy grouping.

The Karnaugh map is a systematic method for simplifying Boolean expressions and, if properly used, will produce the simplest SOP or POS expression possible, known as the minimum expression.

It is similar to a truth table as it presents all of the possible values of input codes and the output for each value. The columns and rows like truth table; the Karnaugh map is an array of cells in which each cell represents a binary value of the input variables. Karnaugh maps can be used for expressionwith 2, 3, 4 or 5 variables. Another method, called the Quine-McClusky method can be used for higher numbers of variables.

For example -functions with five variables in Boolean can be simplified using a 32-cell Karnaugh map. Different types of maps are represented by different blocks or cells like two 4-variable maps (16 cells each) are used to construct a 5-variable map.

**STANDARD FORM OF BOOLEAN EXPRESSION**

1. Sum of Product forms (also known as SOP) - when two or more product terms are added by Boolean addition, the outcome expression is a sum-of-products (SOP). Some examples stated as below-

AB + ABC

ABC + CDE + BCD

AB + BCD + AC

It can be written in a single variable term for example A+ABC+BCD+CDE so on.

**Steps on how to convert Product Term to Standard SOP are stated below-**

All variables in the domain can be expanded to standard SOP including all the variable product terms.Stepsare written to convert a non standard SOP expression to standard SOP form using Boolean algebra technique.

1. Multiplication of each term by term of non standard product made up of the sum of a missing variable and its complement. This results in two product terms as multiplying anything by 1 without changing its value.

2. Repeat Step 1 until all resulting product terms contain all variables in the domain in either complemented or uncomplemented form. In converting a product term to standard form, the number of product terms is doubled for each missing variable.

A summation term was defined as a term consisting of the sum (Boolean addition) of literals (variables or their complements). When two or more sum terms are multiplied, the resulting expression is a product-of-sums (POS). Some examples are

(a+b)(a+b+c)

(a+b+c)(c+d+e)(b+c+d)

(a+b)(a+b+c)(a+c)

A POS expression can contain a single-variable term as a(a+b+c)(b+c+d)

**To construct a truth table from a POS(product of sum) expression.**

1. list all the possible combinations of binary values of the variables just as was done for the SOP expression.

2. convert the POS expression to standard form if it is not already.

3. Put a zero /0 in the output column for each binary value that makes the expression zero/0 and put a 1 for all the remaining binary values.

**COMBINATIONAL LOGIC ANALYSIS**

1.AND-OR Logic- AND-OR circuit consisting of two 2-input AND codes and one 2-input OR gate

2. AND-OR-Invert Logic- When the output of an AND-OR circuit is complemented (inverted); it results in an AND-OR-Invert circuit. Recall that AND-OR logic directly implements SOP expressions. POS expressions can be implemented with AND-OR-Invert logic.

3. Exclusive-OR logic- The exclusive-OR gate was introduced before. Here, circuit is considered a type of logic gate with its own unique symbol and it is actually a combination of two and codes, one or gate, and two inverters.

**Importance of Boolean Algebra**

All our manufacturing companies producing electronic gadgets use binary coding that contains thousand of switches that are arranged into logical codes that produce relevant results.

The basic logic codes are and, or, not. It is these codes, used in different combinations that allow the computer to execute its operations using binary language.

Every logic function assesses information (consisting of high or low voltages) in accordance with defined set of rules, and produces a single, high or low voltage logical conclusion. The voltage codes denotes the binary yes-no, true-false, one-zero concept.

And codes will only yield a TRUE result (that is represented as binary 1) if all input is TRUE, hence the top two codes will produce a FALSE (that is represented binary 0) as result.

Whereas, or codes are less complicated. An or code will send a true result if any of its input is TRUE. Only if every input is false then will it produce a 0 result.

By rewriting the codes, binary logics can be used not only for addition, subtraction, multiplication and division enabling them to ensure smooth functioning of digital instruments like computers data flow and to process various instructions produced as per system.

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