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 Expression Equivalence  Method: Boolean expression equivalence can be proved by either  Applying Boolean Axioms and Theorems or  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 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 1 0 1 1 DIGITAL DESIGN 101, University of California Boolean Algebra

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’ DIGITAL DESIGN 101, University of California Boolean Algebra

Boolean Expression to Canonical Form  Expression  Table  Canonical Form  Expression:  F = xy + x’y’ + xy’z  Create a table for the expression Row Variable Values Output x y z F 0 0 0 0 1 1 0 0 1 1  Use table to create canonical form 2 0 1 0 0  Sum-of-minterms: 3 0 1 1 0  F = Σ (0, 1, 5, 6, 7) 4 1 0 0 0 = x’y’z’ + x’y’z + xy’z + xyz’ + xyz 5 1 0 1 1  Product-of-maxterms 6 1 1 0 1  F = Π (2, 3, 4) 7 1 1 1 1 = (x + y’ + z)(x + y’ + z’)(x’ + y + z) Truth Table DIGITAL DESIGN 101, University of California Boolean Algebra

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 DIGITAL DESIGN 101, University of California Boolean Algebra