Boolean Algebra part 2 (1).pdf
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This document discusses Boolean algebra and its application to simplifying logic circuits. It describes how Boolean expressions can be converted into standard sum-of-products (SOP) or product-of-sums (POS) forms, making them easier to evaluate, simplify, and implement. The standard forms contain all variables in either complemented or uncomplemented terms. Non-standard expressions are converted to standard forms by multiplying or adding missing variable terms until all variables are included in each term.
Boolean Algebra part 2 (1).pdf
- 2. Logic Simplification Using Boolean Algebra • The most practical use of Boolean algebra is to simplify logic circuits. • Using the laws and theorems of Boolean Algebra, the algebraic forms of functions can often be simplified, which leads to simpler (and cheaper) implementations. • This method requires a thorough knowledge of Boolean algebra and considerable practice in its application. • However, there is no easy way to tell whether the simplified expression is in its simplest form or whether it could have been simplified further.
- 3. Boolean expression simplification- example
- 4. Standard Forms of Boolean Expressions • All Boolean expressions, regardless of their form, can be converted into either of two standard forms: ➢The sum-of-products (SOP) form ➢The product-of-sums (POS) form • Standardization makes the evaluation, simplification, and implementation of Boolean expressions much more systematic and easier.
- 5. The Sum-of-Products (SOP) Form • A product term consisting of the product (Boolean multiplication) of literals (variables or their complements). • When two or more product terms are summed by Boolean addition, the resulting expression is a sum-of-products (SOP). • The sum-of-products expression can contain a single variable term. We can have • In an SOP form, a single overbar cannot extend over more than one variable; however, more than one variable in a term can have an overbar. Some examples are
- 6. The Standard SOP Form • If each term in the sum of products form contains all the variables (literals) either in complemented or uncomplemented form , then the expression is known as standard sum of products form or canonical sum of products form. • For example • Each of the product terms in the standard SOP form is called a minterm. The minterms are often denoted as m0 , m1, m2,..., where the subscripts are the decimal equivalent of the binary number of the minterms. • notation is used to represent sum-of-products Boolean expressions.
- 7. The Product-of-Sums (POS) Form • A sum term is 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 of product-of-sums form are • A POS expression can contain a single-variable term. • In a POS expression, a single overbar cannot extend over more than one variable; however, more than one variable in a term can have an overbar. For example
- 8. Standard Product-of-Sums (POS) Form • A standard POS expression is one in which all the variables in the domain appear in each sum term in the expression. For example, • Each term in the standard POS form is called a Maxterm. • Maxterms are often represented as M0 , M1, M2,....,where the subscripts denote decimal equivalent of the binary number of the maxterms. • notation is used to represent product of sums Boolean expressions.
- 9. Conversion of SOP form to Standard SOP The procedure for converting a non-standard sum-of-products expression to a standard form is given by Step 1: Multiply each nonstandard product term by a term made up of the sum of a missing variable and its complement. This results in two product terms. Step 2: Repeat Step 1 until all resulting product terms contain all variables in the domain in either complemented or uncomplemented form. Step 3: Expends the term by applying, distributive law and reorder the literals. Step 4: Reduce the repeated product terms.
- 12. Conversion of POS form to standard POS The procedure for converting a non-standard product-of-sums expression to a standard form. Step 1: Add to each nonstandard product term a term made up of the product of the missing variable and its complement. This results in two sum terms. Step 2: For expanding apply the rule Step 3: Repeat Step 1 until all resulting sum terms contain all variables in the domain in either complemented or uncomplemented form. Step 3: remove the repeated sum terms if any.