1. Computer Organization CS345 David Monismith Based upon notes by Dr. Bill Siever and notes from the Patternson and Hennessy Text

2. Review Last time we reviewed for the exam and in particular we covered Boolean Algebra. This time we will continue Boolean algebra by looking at the sum of products representation and we will look at forming logic gates using JLS from a sum of products.

3. Boolean Algebra A sum of products is a group of Boolean terms that are added (ORed) together where each term is a product (AND) of Boolean variables. For example, the following equation is a sum of products: A*~B*C + ~A*B*~C + A*~B*~C There also exists a product of sums, which is the dual (using DeMorgan's law) of the sum of products. We won't focus much attention on the produce of sums. We can easily convert a truth table to a sum of products and then convert the sum of products to a functioning logic circuit. The algorithm to do so follows below. To do so, examine the following truth table. Look at each row that has a true output (those with a result of 1).

4. Boolean Algebra: Example For each row with a true output, create a product using all input variables. Variables having a false input for a row are negated in the product. Variables having a true input for a row are not negated in the product. Rows with false outputs are ignored when forming a sum of products. A B C Output ------------------------------------ 0 0 0 1 001 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 0 0 1 1

5. Example Contd. Notice that for the input above, there are four rows with true outputs. Those are as follows: A B C Output ------------------------------------ 0 0 0 1 001 1 0 1 1 1 1 1 So, the products for each row, in the order shown above are: ~A*~B*~C ~A*~B*C ~A*B*C A*B*C We then sum (OR) these products to form our sum of products equation as follows: ~A*~B*~C + ~A*~B*C + ~A*B*C + A*B*C

6. Boolean Algebra Notice that when using a sum of products, there are only two levels of gates (aside from the NOT gates). There is one level of ANDs, which occur first, and one level of ORs which occur second. In class, diagrams of AND, OR, and NOT gates were shown. It was also shown that you may have multiple inputs to an AND or an OR gate. Gates like AND, OR, and NOT only have one output, unless the outputs are copied.