Standard Forms of Boolean Expressions q All Boolean

Standard Forms of Boolean Expressions q. All Boolean expressions, regardless of their form, can be converted into either of two standard forms: Ø The Sum-of-products Form or Ø The Product-of-sums Form q. Standardization makes the evaluation, simplification, and implementation of Boolean expressions much more systematic and easier.

4 -6 Standard Forms of Boolean Expressions q

4 -6 Standard Forms of Boolean Expressions q. Domain of a Boolean Expression Ø The domain of a general Boolean expression is the set of variables contained in the expression in either complemented or uncomplemented form. q For example: Ø The domain of the expression: AB + ABC is the set of variables A, B, C Ø The domain of the expression ABC + CDE + BCD is the set of variables A, B, C, D, E.

4 -6 Standard Forms of Boolean Expressions q AND/OR Implementation of an SOP Expression Implementation of the SOP expression AB + BCD + AC

4 -6 Standard Forms of Boolean Expressions q NAND/NAND Implementation of an SOP Expression This NAND/NAND implementation of AB + BCD + AC

4 -6 Standard Forms of Boolean Expressions q. Conversion of a General Expression to SOP Form: Ø Any logic expression can be changed into SOP form by applying Boolean algebra techniques. q. For example Ø The expression A(B + CD) can be converted to SOP form by applying the distributive law: q. A(B + CD) = AB + ACD

4 -6 Standard Forms of Boolean Expressions q

4 -6 Standard Forms of Boolean Expressions q

4 -6 Standard Forms of Boolean Expressions q

4 -6 Standard Forms of Boolean Expressions q

4 -6 Standard Forms of Boolean Expressions q

4 -6 Standard Forms of Boolean Expressions q. An SOP expression is equal to 1 only if one or more of the product terms in the expression is equal to 1. q. Example 4 -16:

4 -6 Standard Forms of Boolean Expressions

Standard Forms of Boolean Expressions q The Product-of-Sums (POS) Form q A sum term is the sum (Boolean addition) of literals (variables or their complements). q When two or more sum terms are multiplied, the resulting expression is a product-of-sums (POS). Some examples are:

4 -6 Standard Forms of Boolean Expressions q The Product-of-Sums (POS) Form q Implementation of a POS Expression Implementation of the POS expression (A + B)(B + C + D)(A + C).

4 -6 Standard Forms of Boolean Expressions q

4 -6 Standard Forms of Boolean Expressions q

4 -6 Standard Forms of Boolean Expressions q Example 5 -17

4 -6 Standard Forms of Boolean Expressions q

4 -6 Standard Forms of Boolean Expressions q Example 4 -18

4 -6 Standard Forms of Boolean Expressions q Converting Standard SOP to Standard POS The binary Evaluate each product term in the SOP expression. That is, determine the binary numbers that represent the product terms. q Step 1: Determine all of the binary numbers not included in the evaluation in Step 1. q Step 2: q Step 3: Write the equivalent sum term for each binary number from Step 2 and express in POS form. q Using a similar procedure, you can go from POS to SOP.

4 -6 Standard Forms of Boolean Expressions q Example 4 -19

4 -7 Boolean Expressions and Truth Tables q Converting SOP Expressions to Truth Table Format q Example 4 -20

4 -7 Boolean Expressions and Truth Tables q Converting POS Expressions to Truth Table Format q Example 4 -21 Notice that the truth table in this example is the same as the one in Example 4– 20. This means that the SOP expression in the previous example and the POS expression in this example are equivalent.

4 -7 Boolean Expressions and Truth Tables q Determining Standard Expressions from a Truth Table: q Example 4 -22

4 -7 Boolean Expressions and Truth Tables