Today’s Agenda

1. Today’s Agenda Boolean Algebra Cannonical and Standard form of Boolean Expression

2. Boolean Algebra • It is a deductive mathematical system that deals with binary variables and logical functions. • Boolean Expressions can be made from binary variables which can have value 0 or 1 and logical operations AND (.), OR (+), NOT (a‘).

3. Postulates and Theorems of Boolean Algebra • x + 0 = x • x + x’ = 1 • x + x = x • x + 1 = 1 • (x’)’ = x • x+y = y+x • x+(y+z) = (x+y)+z • x(y+z) = xy + xz • x+xy = x • x . 1 = x • x . x’ = 0 • x . x = x • x . 0 = 0 • xy = yx • (xy)z = x(yz) • x + (yz) = (x+y)(x+z) • x(x+y) = x

4. Examples • x(x’+y) = xx’ + xy = 0 + xy = xy • x + x’y = (x+x’)(x+y) = 1.(x+y) = (x+y) • (x+y)(x+y’) = x • xy + x’z + yz = xy + x’z • xyz’ + x’yz + xyz + x’yz’ = y

5. Cannonical and Standard form of Boolean functions

6. Cannonical Form • Minterm (or Standard Product) • Maxterms (or Standard Sum)

7. Minterms • A Minterm is a product of all input binary variables in its normal or complement form but not both. • E.g. for 2 variable boolean functions with input x and y, x’y is a minterm but x’ is not a minterm. • E.g for 3 variable boolean functions with input x ,y and z, x’yz is a minterm but x’y is not a minterm.

8. Maxterm (Standard Sum) • Maxterm term is a sum term in boolean expression in which each input variable appears only once in its normal or complement form but not both. • E.g for 3 variable boolean functions with inputs x ,y and z, (x’+y+z) is a maxterm but x’+y is not a minterm.

10. Boolean function can be expressed in Sum of minterms or product of maxterm term. • Given a truth table express the output in sum of minterms and product of maxterm.

12. F0 in Sum of Minterms is • x’y’z’ + x’yz + xyz’ + xyz • F0 = (m0 + m3 + m6 + m7) • F0 = Ʃ (0,3,6,7) • F0 in product of maxterms is • (x+y+z’)(x+y’+z)(x’+y+z)(x’+y+z’) • F0 = M1.M2.M4.M5 • F0 = π(1,2,4,5)

13. Standard Forms • In this configuration, the terms that form a function may contain one , two or any number of literals. • In this form, sum of products is a boolean function containing AND terms called as product terms with one or more literals. • E.g. F1 = y’ + xy + x’yz’ • Similarly product of sums, boolean expression containing OR terms or sum terms with one or more literals • E.g. F2 = x(y’+z)(x’+y+z’)

14. Exercise • Design a circuit to detect whether overflow has occurred or not while adding two 2-bit unsigned binary numbers.