Digital Design Discussion: Boolean Algebra Boolean Expression - - PowerPoint PPT Presentation

Principles Of

Digital Design

Discussion: Boolean Algebra

Boolean Expression Equivalence Boolean Function Complement Boolean Expression to Canonical Form Canonical Form to Standard Form

Boolean Algebra DIGITAL DESIGN 101, University of California

Boolean Expression Equivalence

 Method:

Boolean expression equivalence can be proved by either

Applying Boolean Axioms and Theorems

• r

Checking the truth table

 Example:

Prove equivalence with a truth table

 x + x’y = x + y  Solution: x y x’y x+x’y x + y

1 1 1 1 1 1 1 1 1 1 1

Boolean Algebra DIGITAL DESIGN 101, University of California

Boolean Function Complement

 Problem:

Find the inverse of the following equation and reduce to sum-of-product form:

F = ac' + abd' + acd

 Solution:

F’ = (ac' + abd' + acd )’ // De Morgan’s Law = (ac’)’(abd’)’(acd)’ // De Morgan’s Law = (a’ + c)(a’ + b’ + d)(a’ + c’ + d’) // Distributivity = a’ + c(b’ + d)(c’ + d’) // Commutativity = a’ + c(c’ + d’)(b’ + d) // Distributivity = a’ + (cc’ + cd’)(b’ + d) // cc’ = 0 = a’ + cd’(b’ + d) // Distributivity = a’ + cb’d’ + cdd’ // dd’ = 0 = a’ + cb’d’

Boolean Algebra DIGITAL DESIGN 101, University of California

Boolean Expression to Canonical Form

 Expression  Table  Canonical Form  Expression:

F = xy + x’y’ + xy’z

 Create a table for the expression  Use table to create canonical form

Sum-of-minterms:

• F = Σ(0, 1, 5, 6, 7)

= x’y’z’ + x’y’z + xy’z + xyz’ + xyz

Product-of-maxterms

• F = Π(2, 3, 4)

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

Row Variable Values Output x y z F 1 1 1 1 2 1 3 1 1 4 1 5 1 1 1 6 1 1 1 7 1 1 1 1

Truth Table

Boolean Algebra DIGITAL DESIGN 101, University of California

Canonical Form to Standard Form

 Canonical Form  Find Minterm Factors  Simplify  Expression:

Sum-of-minterms to sum-of-products: F = x’y’z’ + x’y’z + xy’z + xyz’ + xyz = x’y’(z’ + z) + xy’z + xy(z’ + z) // find minterm factors = x’y’ + xy’z + xy // simplify Product-of-maxterms to product-of-sums: F = (x + y’ + z)(x + y’ + z’)(x’ + y + z) = ((x + y’ + z)’ + (x + y’ + z’)’ + (x’ + y + z))’ // De Morgan’s Law = (x’yz’ + x’yz + xy’z’)’ // De Morgan’s Law = (x’y(z’ + z) + xy’z’)’ // find minterm factors = (x’y + xy’z’)’ // simplify = (x’y)’(xy’z’)’ // De Morgan’s Law = (x + y’)(x’ + y + z) // De Morgan’s Law