Department of Computer and Information Science, School of Science, IUPUI

1. Department of Computer and Information Science,School of Science, IUPUI CSCI 240 Boolean Algebra Standard Forms Dale Roberts, Lecturer Computer Science, IUPUI E-mail: droberts@cs.iupui.edu

2. Standard Form • Like regular algebra, Boolean equations can be expressed as a sum of products. Each product is a term of the equation. • Consider the term xyx’z. We define a term to be a Fundamental Product (FP) if it does not repeat any literal. Since our term repeats x, is it not FP. • You can always minimize a term to make it FP.xyx’z = xx’yz = 0yz (which law) = 0 (which law?) • An expression E is sum-of-products form if it is the sum of one or more FPs, none of which is included in another.

3. Sum-of-Products Form 1. Consider E=xz’ + y’z + xyz’. This is a sum of products, but is not sum-of-products form because xz’ is contained within xyz’. i.e. it can be reduced. (which law?) 2. Consider E=xz’ + x’yz’ + xy’z. This is already sum-of-products form. Any non-zero Boolean expression can be changed to sum-of-products form. (By duality, there is also a product-of-sums form, but it is used less often.)

4. Converting to Sum-of-Products Form • Use DeMorgan’s Laws and Involution to move complements inside parenthesis until only variables are complemented. • Use distributive law to transform in to sum of products • Use commutative, idempotent, and complement laws to transform each term into 0 or FP. • Use absorption law to make sum-of-products form. Example • Consider E=((ab)’c)’((a’+c)(b’+c’))’ • E=((ab)’’+c’)((a’+c)’+(b’+c’)’) – DeMorgan’s Law • E=(ab+c’)(ac’+bc) – DeMorgan’s and Involution Laws • E=abac’ + abbc +ac’c’ + bcc’ – Distributive Law • E=abc’ + abc + ac’ + 0 – Idempotent and Complement • E=ac’ + abc – Absorption Law

5. Complete Sum-of-Products Form • A complete sum-of-products form is a sum-of-products form where each term involves all the variables. (Each term will have the same number of literals. • You can add missing variables by multiplying by 1, where 1 is of the form x + x’. • Theorem: Every non-zero Boolean expression can be placed in complete sum-of-products form, and it is unique.

6. Complete Sum-of-Products Example • Express E(x,y,z) = (x’ + y)’ + x’y in complete sum-of-products form. • E = xy’ + x’y – DeMorgan’s Law.If we didn’t know z was involved, we’d think we were done. • E=xy’(z + z’) + x’y(z + z’) • E=xy’z + xy’z’ + x’yz + x’yz’ – Distributive Law(Don’t simplify using Absorption, or you’ll take it back out of Complete S-O-P form)

7. Sources • Lipschutz, Discrete Mathematics • Mowle, A Systematic Approach to Digital Logic Design