Dale Roberts

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

Dale Roberts, LecturerComputer Science, IUPUIE-mail: droberts@cs.iupui.edu

CSCI 240

Boolean AlgebraStandard Forms

Dale Roberts

Standard FormStandard Form

Like regular algebra, Boolean equations can be expressed as a Like regular algebra, Boolean equations can be expressed as a sum of products. Each product is a sum of products. Each product is a termterm of the equation. of the equation.

Consider the term xyx’z. We define a term to be a Consider the term xyx’z. We define a term to be a Fundamental ProductFundamental Product (FP) if it does not repeat any literal. (FP) if it does not repeat any literal. Since our term repeats x, is it not FP.Since our term repeats x, is it not FP.

You can always minimize a term to make it FP.You can always minimize a term to make it FP.xyx’z = xx’yz = 0yz (which law) = 0 (which law?)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 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.or more FPs, none of which is included in another.

Dale Roberts

Sum-of-Products FormSum-of-Products Form1. 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.)

Dale Roberts

Converting to Sum-of-Products FormConverting to Sum-of-Products Form1.1. Use DeMorgan’s Laws and Involution to move complements Use DeMorgan’s Laws and Involution to move complements

inside parenthesis until only variables are complemented.inside parenthesis until only variables are complemented.

2.2. Use distributive law to transform in to sum of productsUse distributive law to transform in to sum of products

3.3. Use commutative, idempotent, and complement laws to transform Use commutative, idempotent, and complement laws to transform each term into 0 or FP.each term into 0 or FP.

4.4. Use absorption law to make sum-of-products form.Use absorption law to make sum-of-products form.

ExampleExampleConsider Consider E=((ab)’c)’((a’+c)(b’+c’))’E=((ab)’c)’((a’+c)(b’+c’))’

E=((ab)’’+c’)((a’+c)’+(b’+c’)’) – DeMorgan’s LawE=((ab)’’+c’)((a’+c)’+(b’+c’)’) – DeMorgan’s Law

E=(ab+c’)(ac’+bc) – DeMorgan’s and Involution LawsE=(ab+c’)(ac’+bc) – DeMorgan’s and Involution Laws

E=abac’ + abbc +ac’c’ + bcc’ – Distributive LawE=abac’ + abbc +ac’c’ + bcc’ – Distributive Law

E=abc’ + abc + ac’ + 0 – Idempotent and ComplementE=abc’ + abc + ac’ + 0 – Idempotent and Complement

E=ac’ + abc – Absorption LawE=ac’ + abc – Absorption Law

Dale Roberts

Complete Sum-of-Products FormComplete Sum-of-Products Form

A A complete sum-of-productscomplete sum-of-products form is a sum-of-products form is a sum-of-products form where each term involves all the variables. form where each term involves all the variables. (Each term will have the same number of literals.(Each term will have the same number of literals.

You can add missing variables by multiplying by 1, You can add missing variables by multiplying by 1, where 1 is of the form where 1 is of the form x + x’.x + x’.

Theorem: Every non-zero Boolean expression can be Theorem: Every non-zero Boolean expression can be placed in complete sum-of-products form, and it is placed in complete sum-of-products form, and it is uniqueunique..

Dale Roberts

Complete Sum-of-Products ExampleComplete Sum-of-Products ExampleExpress Express E(x,y,z) = (x’ + y)’ + x’yE(x,y,z) = (x’ + y)’ + x’y in complete sum-of- in complete sum-of-products form.products form.

E = xy’ + x’yE = xy’ + x’y – DeMorgan’s Law. – DeMorgan’s Law.If we didn’t know z was involved, we’d think we were If we didn’t know z was involved, we’d think we were done.done.

E=xy’(z + z’) + x’y(z + z’)E=xy’(z + z’) + x’y(z + z’)

E=xy’z + xy’z’ + x’yz + x’yz’E=xy’z + xy’z’ + x’yz + x’yz’ – Distributive Law – Distributive Law(Don’t simplify using Absorption, or you’ll take it back (Don’t simplify using Absorption, or you’ll take it back out of Complete S-O-P form)out of Complete S-O-P form)

Dale Roberts

SourcesSources

Lipschutz, Discrete Mathematics

Mowle, A Systematic Approach to Digital Logic Design