1. Chapter 2

2. Outlines 2.1 Introduction 2.2 Basic Definitions 2.3 Axiomatic Definition of Boolean Algebra 2.4 Basic thermos and proprieties of Boolean Algebra 2.5 Boolean Functions 2.6 Canonical and standard forms 2.7 Other Logic Operations 2.8 Digital Logic Gates

3. Introduction Because binary logic is used in all of today’s digital computers and devices, the cost of the circuits that implement it is an important factor addressed by designers. This chapter provides a basic vocabulary and a brief foundation in Boolean algebra that will enable you to optimize simple circuits and to understand the purpose of algorithms used by software tools to optimize complex circuits involving millions of logic gates.

4. Basic Definitions of Algebra Algebra Basic definitions: 1. Closure: A set Sis closed with respective to a binary operator if, for every pair of elements of S, the binary operator specifies a rule for obtaining a unique element of S. 2. Associative law: A binary operator * on a set S is said to be associative whenever (x*y)*z= x*(y*z) for all x, y, z ∈ S. 3. Communicative law: A binary operator * on a set Sis said to be communicative whenever x*y= y*x for all x, y ∈ S. 4. Identity element: A set Sis said to have an identity element with respect to a binary operation * on S if there exists an element e ∈ S with the property that x*e = e*x =x for every x ∈ S. 5. Inverse: A set Shaving the identity element e with respect to a binary operation * is said to have an inverse whenever x ∈ S, there exists an element y ∈ S such that x*y = e. 6. Distributive law: If * and ‧ are two binary operators on a set S, * is said to be distributive over ‧ whenever x*(y ‧ z)=(x*y) ‧ (x*z)

5. Axiomatic Definition of Boolean Algebra Boolean Algebra defined by a set of elements B and two binary operators + and ‧ and has the following postulates: Postulate 1: (a) The structure is closed with respect to the operator +. (b) The structure is closed with respect to the operator ‧. Postulate 2: (a) The element 0 is an identity element with respect to +. 0+x = x+0 = x (b) The element 1 is an identity element with respect to ‧. 1.x = x ‧ 1= x Postulate 3: (a) The structure is communicative with respect to +. x + y = y + x (b) The structure is communicative with respect to ‧. X ‧ y = y ‧ x

6. Axiomatic Definition of Boolean Algebra Postulate 4 (a) The operator ‧ is distributive over +: x ‧ (y+z)=(x ‧ y)+(x ‧ z) (b) The operator + is distributive over ‧ x+(y ‧ z)=(x+y) ‧ (x+z) Postulate 5: (a) For every element x ∈ B, there exists an element x' ∈ B (complement of x) such that (a)x+x'=1 and (b) x ‧ x'=0. Postulate 6: There exist at least two elements x, y ∈ B such that x y. Difference with ordinary algebra The operator + is distributive over ‧ is valid for Boolean algebra, but not for ordinary algebra. Boolean algebra does not have additive and multiplicative inverses; therefore, there are not subtraction or division operations. Complement is valid for Boolean algebra, but not for ordinary algebra.

7. Two-Valued Boolean Algebra Postulate 4: The distributive law can be shown to hold from the truth table of all possible values of x, y, and z. Postulate 5: Complement x+x'=1: since 0+0'=0+1=1 and 1+1'=1+0=1 x ‧ x'=0: since 0 ‧ 0'=0 ‧ 1=0 and 1 ‧ 1'=1 ‧ 0=0 Postulate 6: Has two distinct elements 1 and 0, with 0 1 Note A set of two elements + : OR operation; ‧ : AND operation A complement operator: NOT operation Binary logic is a two-valued Boolean algebra

8. Basic Theorems and Properties Duality The binary operators are interchanged; AND ⇔ OR The identity elements are interchanged; 1 ⇔ 0

9. Basic Theorems and Properties Theorem 1(a): x+x = x –x+x = (x+x) 1by postulate 2(b)= (x+x) (x+x')5(a)= x+xx'4(b)= x+05(b)= x2(a) –Theorem 1(b): x ‧ x= x –x ‧ x= x ‧ x + 0by postulate 2(a) = xx + xx'5(b)= x (x + x')4(a)= x ‧ 15(a)= x2(b)

10. Basic Theorems and Properties Theorem 2(a): x + 1 = 1 x + 1 = 1 (x + 1)= (x + x')(x +1)= x + x'1= x + x‘= 1 Theorem 2(b): x ‧ 0 =0 By duality Theorem 3: (x')'= x Postulate 5 defines the complement of x, x + x' = 1 and x ‧ x' = 0 he complement of x' is x is also (x')'

11. Basic Theorems and Properties Theorem 6 (a): x + xy= x x + xy= x 1 + xy= x (1 +y) 1.x= x ‧ 1= x Theorem 6 (b): x(x+y) = x By duality By means of truth table x y xy x + xy

12. Basic Theorems and Properties

13. Boolean Functions A Boolean function binary variables binary operators OR and AND operator NOT Examples F1= x y z‘ F2= x + y'z F3 = x' y' z + x' y z + x y‘ F4= x y' + x' z

14. Boolean Functions

15. Implementation with logic gatesF4 is more economical F3 F2 F4

16. Algebraic Manipulation To minimize Boolean expressions literal: a primed or unprimed variable (an input to a gate) term: an implementation with a gate The minimization of the number of literals and the number of terms => a circuit with less equipment It is a hard problem (no specific rules to follow) Ex1: x(x'+y) = xx' + xy= 0+ xy= xy Ex2: x+x'y= (x+x')(x+y) = 1 (x+y) = x+y Ex3: (x+y)(x+y') = x+xy+xy'+yy' = x(1+y+y') = x Ex4: x'y'z+ x'yz+ xy' = x'z(y'+y) + xy'= x'z+ xy‘ Ex5: xy+ x'z+ yz= xy+ x'z+ yz(x+x')= xy+ x'z+ yzx+ yzx'= xy(1+z) + x'z(1+y) = xy+x'z Ex6: (x+y)(x'+z)(y+z) = (x+y)(x'+z) by duality from the previous result

17. Complement of a Function

19. Canonical and Standard Forms

21. Canonical Forms

25. Conversion between Two Canonical Forms

27. Standard Forms

29. Other Logic Operations

31. Digital Logic Gates

38. Positive and Negative Logic