CHAPTER 2 Boolean Algebra This chapter in the book includes: Objectives Study Guide 2.1 Introduction 2.2 Basic Operation 2.3 Boolean Expression and Truth Table 2.4 Basic Theorem 2.5 Commutative, Associative and Distributive Laws 2.6 Simplification Theorem 2.7 Multiplying Out and Factoring 2.8 DeMorgan’s Laws Problems Laws and Theorems of Boolean Algebra Fundamentals of Logic Design Chap. 2

Objectives Topics introduced in this chapter: Understand the basic operations and laws of Boolean algebra Relate these operations and laws to AND, OR, NOT gates and switches Prove these laws using a truth table Manipulation of algebraic expression using - Multiplying out - Factoring - Simplifying - Finding the complement of an expression Fundamentals of Logic Design Chap. 2

2.1 Introduction Basic mathematics for logic design: Boolean algebra Restrict to switching circuits( Two state values 0, 1) – Switching algebra Boolean Variable : X, Y, … can only have two state values (0, 1) representing True(1) False (0) Fundamentals of Logic Design Chap. 2

2.2 Basic Operations NOT(Inverter) Gate Symbol Fundamentals of Logic Design Chap. 2

2.2 Basic Operations AND Truth Table Gate Symbol Fundamentals of Logic Design Chap. 2

2.2 Basic Operations OR Truth Table Gate Symbol Fundamentals of Logic Design Chap. 2

2.2 Basic Operations Apply to Switch AND OR Fundamentals of Logic Design Chap. 2

2.3 Boolean Expressions and Truth Tables Logic Expression : Circuit of logic gates : Fundamentals of Logic Design Chap. 2

2.3 Boolean Expressions and Truth Tables Logic Expression : Circuit of logic gates : Logic Evaluation : A=B=C=1, D=E=0 Literal : a variable or its complement in a logic expression 10 literals Fundamentals of Logic Design Chap. 2

2.3 Boolean Expressions and Truth Tables 2-Input Circuit and Truth Table A B A’ F = A’ + B 0 0 0 1 1 0 1 1 1 Fundamentals of Logic Design Chap. 2

2.3 Boolean Expressions and Truth Tables Proof using Truth Table n times n variable needs rows TABLE 2.1 Fundamentals of Logic Design Chap. 2

2.4 Basic Theorems Operations with 0, 1 Idempotent Laws Involution Laws Complementary Laws Proof Example Fundamentals of Logic Design Chap. 2

2.4 Basic Theorems with Switch Circuits Fundamentals of Logic Design Chap. 2

2.4 Basic Theorems with Switch Circuits Fundamentals of Logic Design Chap. 2

2.4 Basic Theorems with Switch Circuits Fundamentals of Logic Design Chap. 2

2.5 Commutative, Associative, and Distributive Laws Commutative Laws: Associative Laws: Proof of Associate Law for AND Fundamentals of Logic Design Chap. 2

Associative Laws for AND and OR Figure 2-3: Associative Law for AND and OR Fundamentals of Logic Design Chap. 2

2.5 Commutative, Associative, and Distributive Laws OR Distributive Laws: Valid only Boolean algebra not for ordinary algebra Proof Fundamentals of Logic Design Chap. 2

2.6 Simplification Theorems Useful Theorems for Simplification Proof Proof with Switch Fundamentals of Logic Design Chap. 2

2.6 Simplification Theorems Equivalent Gate Circuits === Figure 2-4: Equivalent Gate Circuits Fundamentals of Logic Design Chap. 2

2.6 Simplification Theorems Simplify (p. 43-44) Fundamentals of Logic Design Chap. 2

2.7 Multiplying Out and Factoring To obtain a sum-of-product form  Multiplying out using distributive laws Sum of product form: Still considered to be in sum of product form: Not in Sum of product form: Multiplying out and eliminating redundant terms Fundamentals of Logic Design Chap. 2

2.7 Multiplying Out and Factoring To obtain a product of sum form  all sums are the sum of single variable Product of sum form: Still considered to be in product of sum form: Not in Product of sum form : Fundamentals of Logic Design Chap. 2

2.7 Multiplying Out and Factoring EXAMPLE 1: Factor A + B'CD. This is of the form X + YZ where X = A, Y = B', and Z = CD, so A + B'CD = (X + Y)(X + Z) = (A + B')(A + CD)   A + CD can be factored again using the second distributive law, so A + B'CD = (A + B')(A + C)(A + D) EXAMPLE 2: Factor AB' + C'D Fundamentals of Logic Design Chap. 2

2.7 Multiplying Out and Factoring EXAMPLE 3: Factor C'D + C'E' + G'H Fundamentals of Logic Design Chap. 2

Circuits for SOP and POS form Sum of product form: Product of sum form: Fundamentals of Logic Design Chap. 2

2.8 DeMorgan’s Laws DeMorgan’s Laws Proof X Y X’ Y’ X + Y ( X + Y )’ XY ( XY )’ X’ + Y’ 0 0 0 1 1 0 1 1 1 DeMorgan’s Laws for n variables Example Fundamentals of Logic Design Chap. 2

2.8 DeMorgan’s Laws Fundamentals of Logic Design Chap. 2

2.8 DeMorgan’s Laws Inverse of A’B+AB’ 0 0 0 1 1 0 1 1 1 Dual: ‘dual’ is formed by replacing AND with OR, OR with AND, 0 with 1, 1 with 0 Fundamentals of Logic Design Chap. 2

LAWS AND THEOREMS (a) Operations with 0 and 1: 1. X + 0 = X 1D. X • 1 = X 2. X +1 = 1 2D. X • 0 = 0   Idempotent laws: 3. X + X = X 3D. X • X = X Involution law: 4. (X')' = X Laws of complementarity: 5. X + X' = 1 5D. X • X' = 0 Fundamentals of Logic Design Chap. 2

LAWS AND THEOREMS (b) Commutative laws: 6. X + Y = Y + X 6D. XY = YX   Associative laws: 7. (X + Y) + Z = X + (Y + Z) 7D. (XY)Z = X(YZ) = XYZ = X + Y + Z Distributive laws: 8. X(Y + Z) = XY + XZ 8D. X + YZ = (X + Y)(X + Z) Simplification theorems: 9. XY + XY' = X 9D. (X + Y)(X + Y') = X 10. X + XY = X 10D. X(X + Y) = X 11. (X + Y')Y = XY 11D. XY' + Y = X + Y Fundamentals of Logic Design Chap. 2

LAWS AND THEOREMS (c) DeMorgan's laws: 12. (X + Y + Z +...)' = X'Y'Z'... 12D. (XYZ...)' = X' + Y' + Z' +...   Duality: 13. (X + Y + Z +...)D = XYZ... 13D. (XYZ...)D = X + Y + Z +... Theorem for multiplying out and factoring: 14. (X + Y)(X' + Z) = XZ + X'Y 14D. XY + X'Z = (X + Z)(X' + Y) Consensus theorem: 15. XY + YZ + X'Z = XY + X'Z 15D. (X + Y)(Y + Z)(X' + Z) = (X + Y)(X' + Z) Fundamentals of Logic Design Chap. 2