Boolean Algebra and Logic Gates COE 202 Digital

Boolean Algebra and Logic Gates COE 202 Digital Logic Design Dr. Muhamed Mudawar King Fahd University of Petroleum and Minerals

Presentation Outline v Boolean Algebra v Boolean Functions and Truth Tables v De. Morgan's Theorem v Algebraic manipulation and expression simplification v Logic gates and logic diagrams v Minterms and Maxterms v Sum-Of-Products and Product-Of-Sums Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 2

Boolean Algebra v Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 3

Postulates of Boolean Algebra v Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 4

AND, OR, and NOT Operators v x y x·y x y x+y x x' 0 0 0 0 1 1 1 0 0 1 1 1 1 Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 5

Boolean Functions v Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 6

Truth Table v x y z y' x' x'z f = xy'+ x'z 0 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 7

De. Morgan's Theorem v Can be verified Using a Truth Table x y x' y' x+y (x+y)' x'y' x y (x y)' x'+ y' 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 0 Identical Boolean Algebra and Logic Gates COE 202 – Digital Logic Design Identical © Muhamed Mudawar – slide 8

Complementing Boolean Functions v Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 9

Algebraic Manipulation of Expressions v Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 10

Consensus Theorem v Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 11

Summary of Boolean Algebra Property Dual Property Identity Complement Null Idempotence Involution Commutative Associative Distributive Absorption Simplification De Morgan Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 12

Duality Principle v Property Dual Property Identity Complement Distributive Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 13

Expression Simplification v Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 14

Importance of Boolean Algebra v Our objective is to learn how to design digital circuits v These circuits use signals with two possible values v Logic 0 is a low voltage signal (around 0 volts) v Logic 1 is a high voltage signal (e. g. 5 or 3. 3 volts) v The physical value of a signal is the actual voltage it carries, while its logic value is either 0 (low) or 1 (high) v Having only two logic values (0 and 1) simplifies the implementation of the digital circuit Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 15

Next. . . v Boolean Algebra v Boolean Functions and Truth Tables v De. Morgan's Theorem v Algebraic manipulation and expression simplification v Logic gates and logic diagrams v Minterms and Maxterms v Sum-Of-Products and Product-Of-Sums Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 16

Logic Gates and Symbols AND gate AND: Switches in series logic 0 is open switch OR gate OR: Switches in parallel logic 0 is open switch NOT gate (inverter) NOT: Switch is normally closed when x is 0 v In the earliest computers, relays were used as mechanical switches controlled by electricity (coils) v Today, tiny transistors are used as electronic switches that implement the logic gates (CMOS technology) Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 17

Truth Table and Logic Diagram v Truth Table Logic Diagram x y z y'+ z f = x(y'+ z) 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 1 1 1 Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 18

Combinational Circuit v Boolean Algebra and Logic Gates Circuit COE 202 – Digital Logic Design Combinational © Muhamed Mudawar – slide 19

Example of a Simple Combinational Circuit v Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 20

From Truth Tables to Gate Implementation v Truth Table x y z f 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 Boolean Algebra and Logic Gates To answer these questions, we need to define Minterms and Maxterms COE 202 – Digital Logic Design © Muhamed Mudawar – slide 21

Minterms and Maxterms v x y index 0 0 1 1 1 0 2 1 1 3 Boolean Algebra and Logic Gates Minterm COE 202 – Digital Logic Design Maxterm © Muhamed Mudawar – slide 22

Minterms and Maxterms for 3 Variables x y z index 0 0 0 1 1 0 2 0 1 1 3 1 0 0 4 1 0 1 5 1 1 0 6 1 1 1 7 Minterm Maxterm v Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 23

Purpose of the Index v Minterms and Maxterms are designated with an index v The index for the Minterm or Maxterm, expressed as a binary number, is used to determine whether the variable is shown in the true or complemented form v For Minterms: ² ‘ 1’ means the variable is Not Complemented ² ‘ 0’ means the variable is Complemented v For Maxterms: ² ‘ 0’ means the variable is Not Complemented ² ‘ 1’ means the variable is Complemented Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 24

Sum-Of-Minterms (SOM) Canonical Form Truth Table x y z f 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 Boolean Algebra and Logic Gates Minterm Sum of Minterm entries that evaluate to ‘ 1’ Focus on the ‘ 1’ entries COE 202 – Digital Logic Design © Muhamed Mudawar – slide 25

Examples of Sum-Of-Minterms v Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 26

Product-Of-Maxterms (POM) Canonical Form Truth Table x y z f 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 Boolean Algebra and Logic Gates Maxterm Product of Maxterm entries that evaluate to ‘ 0’ Focus on the ‘ 0’ entries COE 202 – Digital Logic Design © Muhamed Mudawar – slide 27

Examples of Product-Of-Maxterms v Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 28

Conversions between Canonical Forms v Truth Table x y z f 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 Boolean Algebra and Logic Gates Minterms Maxterms To convert from one canonical form to another, interchange the symbols and list those numbers missing from the original form. COE 202 – Digital Logic Design © Muhamed Mudawar – slide 29

Function Complement v Truth Table x y z f f' 0 0 0 1 0 1 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 Boolean Algebra and Logic Gates The complement of a function expressed by a Sum of Minterms is the Product of Maxterms with the same indices. Interchange the symbols and , but keep the same list of indices. COE 202 – Digital Logic Design © Muhamed Mudawar – slide 30

Summary of Minterms and Maxterms v There are 2 n Minterms and Maxterms for Boolean functions with n variables, indexed from 0 to 2 n – 1 v Minterms correspond to the 1 -entries of the function v Maxterms correspond to the 0 -entries of the function v Any Boolean function can be expressed as a Sum-of-Minterms and as a Product-of-Maxterms v For a Boolean function, given the list of Minterm indices one can determine the list of Maxterms indices (and vice versa) v The complement of a Sum-of-Minterms is a Product-of-Maxterms with the same indices (and vice versa) Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 31

Sum-of-Products and Products-of-Sums v Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 32

Two-Level Gate Implementation Boolean Algebra and Logic Gates AND-OR implementations 3 -input AND gate OR-AND implementations 3 -input OR gate COE 202 – Digital Logic Design © Muhamed Mudawar – slide 33

Two-Level vs. Three-Level Implementation v 3 -input OR gate Boolean Algebra and Logic Gates COE 202 – Digital Logic Design © Muhamed Mudawar – slide 34