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Understanding Boolean Algebra: Canonical Forms and Conversion Methods

Dive into the world of Boolean algebra, exploring canonical forms and conversion techniques for simplifying expressions. Learn about sum-of-products and product-of-sums forms, logical equivalence, and how to convert functions using truth tables.

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Understanding Boolean Algebra: Canonical Forms and Conversion Methods

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  1. CS 3501 - Chapter 3 (3A and 10.2.2) Dr. Clincy Professor of CS • Today: Brief lecture • Today: Review Exam 1 Dr. Clincy Lecture Slide 1

  2. Consensus Theorem F(x,y,z) = xy + x′z + yz

  3. Consensus Theorem Working backwards and adding a term

  4. Boolean Algebra • Through our exercises in simplifying Boolean expressions, we see that there are numerous ways of stating the same Boolean expression. • These “synonymous” forms are logically equivalent. • Logically equivalent expressions have identical truth tables. • In order to eliminate as much confusion as possible, designers express Boolean functions in standardized or canonical form. Lecture

  5. Boolean Algebra • There are two canonical forms for Boolean expressions: sum-of-products and product-of-sums. • Recall the Boolean product is the AND operation and the Boolean sum is the OR operation. • In the sum-of-products form, ANDed variables are ORed together. • For example: • In the product-of-sums form, ORed variables are ANDed together: • For example: Lecture

  6. Boolean Algebra • It is easy to convert a function to sum-of-products form using its truth table. • We are interested in the values of the variables that make the function true (=1). • Using the truth table, we list the values of the variables that result in a true function value. • Each group of variables is then ORed together. • The sum-of-products form for our function is: We note that this function is not in simplest terms. Our aim is only to rewrite our function in canonical sum-of-products form. Lecture

  7. Boolean Algebra • It is easy to convert a function to product-of-sums form. • We are interested in the values of the variables that make the function true (=0). • Using the truth table, we list the values of the variables that result in a false function value. • Each group of variables is then ANDed together. • The product-of-sum form for our function is: f(x,y,z)=(x+y+z)(x+y+z’)(x’+y+z’) Lecture

  8. CS3503 Exam 1 Results – 2PM Average Score = 31 (Average Grade = 75) Score SD = 19 (large) Grading Scaled Used: 80-61 A-grade (1 student) 60-41 B-grade (13 students) 40-21 C-grade (11 students) 20-1 D-grade (11 students) 0 F-grade (2 students) In getting your grade logged, be sure and pass back the exam after we go over them – Exam Policy – lose points for not passing back Dr. Clincy 8

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