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

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

2.1 Introduction Digital Circuits: being consist of transistors, diodes, resistors, hardware functional elements to process binary (0,1) data. ->Each basic circuit is called as a gate processing a logical operation(AND, OR,NOT,..). Each gate has one or more input ports and one output port (sometime with its complement). Boolean algebra: mathematical basis to specify the operations of gates. It is used to describe the inter-connections of digital gates and circuit diagrams. ->Switching algebra, two-valued Boolean algebra, or Binarly logic is a special case of Boolean algebra. It has two state values, 0 and 1.

George Boole (1858-1908)

2.1 Introduction we are assuming ‘two-valued Boolean algebra’. Boolean variable or binary variable : A, B, X, Y, … can only have one of two state values (0, 1) - representing True(1) or False (0) positive logic: 1 for High, 0 for Low negative logic: 1 for Low, 0for High *In TTL, data sheet may say, High: above 2.7 V, Low: below 0.8 V, undefined in the middle.

2.2 Basic Operations Boolean Operation, NOT: ‘ Circle indicates inversion. Gate Symbol: NOT gate, or Inverter

2.2 Basic Operations A B C = A · B 0 0 0 1 1 0 1 1 1 AND: . or nothing *Note that it is not an arithmetic but a Boolean operation. A B C = A · B 0 0 0 1 1 0 1 1 1 Truth Table *Also known as logical product. Logic Gate Symbol

2.2 Basic Operations A B C = A + B 0 0 0 1 1 0 1 1 1 OR: + Truth Table 0 0 0 1 1 0 1 1 1 Truth Table *Also known as logical summation. Gate Symbol

2.2 Basic Operations Apply to Switch AND OR

2.3 Boolean Expressions and Truth Tables Boolean Expression : basic operations to binary variables, constants (0, 1), logic operation symbols, ( ..), and/or equal sign (=). It is also used to show input to output relationship of binary variables. Priority: parentheses -> NOT -> AND -> OR Circuit of logic gates : being corresponded to each Boolean expression D=

2.3 Boolean Expressions and Truth Tables Logic Expression : Circuit of logic gates : Y= Logic Evaluation : substituting a value to each variable as A=B=C=1, D=E=0 Literal : a variable or its complement in a logic expression 10 literals, 4 terms

2-Input Circuit and Truth Table 2.3 Boolean Expressions and Truth Tables 2-Input Circuit and Truth Table Truth Table: specifies the output of the circuit for all possible combinations of input values. A B A’ F = A’ + B 0 0 0 1 1 0 1 1 1

2.3 Boolean Expressions and Truth Tables Proof using Truth Table: two expressions are equal if they have the same output values for every possible input combinations. n times n variables need rows TABLE 2.1 A B C B’ AB’ AB’ + C A + C B’ + C (A + C) (B’ + C) 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 1

2.4 Basic Theorems Operations with 0, 1 Idempotent Laws Involution Laws Complementary Laws Proof Example

2.4 Basic Theorems Duality: The dual of an algebraic expression is obtained by replacing OR to AND, AND to OR, 1 to 0, and 0 to 1. If there is a valid theorem, then its dual is also a valid theorem. (* Variables are not changed and operational priority must be kept.) EX: A + B* C + 0 => A* (B+C)*1

2.5 Commutative, Associative, and Distributive Laws Some useful laws applied in the Boolean algebra. Commutative Laws: ordering of variables is not important in the definition of AND and OR operations. Associative Laws: among the three variables forming the AND or OR operation, the result is independent of which pair of variables is associated first. So, parentheses can be omitted. This requires hardware attentions, too.

Associative Laws for AND and OR Replacement to 3-input gates

2.5 Commutative, Associative, and Distributive Laws Multiple AND: Multiple OR: Distributive Laws: Valid only Boolean algebra not for ordinary algebra OR also can be distributed over AND. Proof

2.6 Simplification Theorems Useful Theorems for Simplification Proof NOTE: the others are valid as the duals, simplifying the expression results in the reduction of hardware circuits. The first two theorems are the basics for further simplification methods.

Equivalent Gate Circuits 2.6 Simplification Theorems Equivalent Gate Circuits === *Read some examples in the text.

2.7 Multiplying Out and Factoring To obtain a sum-of-products form  Multiplying out using distributive laws Sum-of-products form: sum of product terms consisting of single variables Still considered to be in sum of product form: not a single variable Not in Sum of product form: *Multiplying out and eliminating redundant terms to get s-o-p forms:

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: *These two make the hardware logic circuits as the two-level resulting in the same response times. Sum of Products: OR of AND’s Product of Sums: AND of OR’s

Circuits for SOP and POS form Sum of product form: Product of sum form:

2.8 DeMorgan’s Laws DeMorgan’s Laws: to obtain the inverse Proof X Y X’ Y’ X + Y ( X + Y )’ X’ Y’ XY ( XY )’ X’ + Y’ 0 0 0 1 1 0 1 1 1 1 1 0 0 1 0 0 1 Extension: DeMorgan’s Laws for n variables Example

expression and then complementing each individual variable. 2.8 DeMorgan’s Laws Inverse of A’B+AB’ A B A’ B A B’ F = A’B+AB’ A’ B’ A B F’ = 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 note: variables are not inverted. The complement of an expression may be found by making a dual of the entire expression and then complementing each individual variable.

MOS and CMOS MOSFET (metal-oxide-semiconductor field-effect transistor) C(Complementary) MOS General MOSFET symbol VDD = Power supply voltage, n channel: positive logic. logic 0 = 0 volt and logic 1 = VDD volt p channel: negative logic, logic 0 = 0 volt and logic 1 = -Vnn volt. VDS=negative logic 1 to gate will switch the MOSFET to the ON

MOS Inverter P611, Fig. A.2 For n channel, positive logic case.

TTL(Transistor Transistor Logic) Ref. Digital Electronics, by Uffenbeck Bipolar Transistor: Base, Emitter, Collector TR state: cut-off, active, saturation Vcc Base Collector Emitter

Transistor-Transistor-Logic: Base, Emitter, Collector off off

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When a high voltage (>2 When a high voltage (>2.7) is applied to input, input current is allowed up to 20 uA. When a low voltage (<0.4) is applied to input, input current is allowed up to 0.4 mA.

Homework 6th Ed. 1, 2, 6, 7,12, 22, 25, 26, 29