Boolean Algebra and Logic Gates 1 Computer Engineering (Logic Circuits) Lec. # 3 Dr. Tamer Samy Gaafar Dept. of Computer & Systems Engineering Faculty of Engineering Zagazig University

Boolean Algebra and Logic Gates Course Web Page

Announcements  30 Minutes Quiz will be held at the beginning of lecture.  No Cheatings.  Zero marks for the cheaters.  Do not forget to write your names on the answer sheet. Boolean Algebra and Logic Gates 3

Lec. # 3 Binary Logic & Gates Boolean Algebra and Logic Gates 4

Boolean Algebra and Logic Gates 5 Binary Logic and Gates  Binary variables take on one of two values.  Logical operators operate on binary values and binary variables.  Basic logical operators are the logic functions AND, OR and NOT.  Logic gates implement logic functions.  Boolean Algebra: a useful mathematical system for specifying and transforming logic functions.  We study Boolean algebra as a foundation for designing and analyzing digital systems!

Boolean Algebra and Logic Gates 6 Binary Variables  Recall that the two binary values have different names: True/False On/Off Yes/No 1/0  We use 1 and 0 to denote the two values.  Variable identifier examples: A, B, y, z, or X 1 for now RESET, START_IT, or ADD1 later

Boolean Algebra and Logic Gates 7 Logical Operations  The three basic logical operations are: AND OR NOT  AND is denoted by a dot (·).  OR is denoted by a plus (+).  NOT is denoted by an over bar ( ¯ ), a single quote mark (') after.

Boolean Algebra and Logic Gates 8  Examples: is read “Y is equal to A AND B.” is read “z is equal to x OR y.” is read “X is equal to NOT A.” Notation Examples  Note: The statement: = 2 (read “one plus one equals two”) is not the same as = 1 (read “1 or 1 equals 1”).  BAY  yxz   AX 

Boolean Algebra and Logic Gates 9 Operator Definitions  Operations are defined on the values "0" and "1" for each operator: AND 0 · 0 = 0 0 · 1 = 0 1 · 0 = 0 1 · 1 = 1 OR = = = = 1 NOT 10  01  Buffer  1  1 00

Boolean Algebra and Logic Gates X NOT XZ  Truth Tables  Tabular listing of the values of a function for all possible combinations of values on its arguments  Example: Truth tables for the basic logic operations: Z = X·Y YX AND OR XYZ = X+Y

Filling A Truth Table ( i/ps) Boolean Algebra and Logic Gates 11

Boolean Algebra and Logic Gates 12 dcba

Boolean Algebra and Logic Gates 13  Using Switches Inputs:  logic 1 is switch closed  logic 0 is switch open Outputs:  logic 1 is light on  logic 0 is light off. NOT input:  logic 1 is switch open  logic 0 is switch closed Logic Function Implementation Switches in series => AND Switches in parallel => OR C Normally-closed switch => NOT

Boolean Algebra and Logic Gates 14  Example: Logic Using Switches  Light is on (L = 1) for L(A, B, C, D) = and off (L = 0), otherwise.  Useful model for relay and CMOS gate circuits, the foundation of current digital logic circuits Logic Function Implementation – cont’d B A D C A (B C + D) = A B C + A D

Boolean Algebra and Logic Gates 15 Logic Gates  In the earliest computers, switches were opened and closed by magnetic fields produced by energizing coils in relays. The switches in turn opened and closed the current paths.  Later, vacuum tubes that open and close current paths electronically replaced relays.  Today, transistors are used as electronic switches that open and close current paths.

Boolean Algebra and Logic Gates 16 Logic Gate Symbols and Behavior  Logic gates have special symbols:  And waveform behavior in time as follows : X 0011 Y0101 X · Y(AND)0001 X+ Y(OR)0111 (NOT)X 1100 OR gate X Y Z= X+ Y X Y Z= X · Y AND gate X Z= X NOT gate or inverter YY BUFFER

Boolean Algebra and Logic Gates 17

Boolean Algebra and Logic Gates 18 `

Boolean Algebra and Logic Gates 19 Truth Tables – Cont’d  Used to evaluate any logic function  Consider F(X, Y, Z) = X Y + Y Z XYZX YYY ZF = X Y + Y Z

Boolean Algebra and Logic Gates 20 Logic Diagrams and Expressions  Boolean equations, truth tables and logic diagrams describe the same function!  Truth tables are unique, but expressions and logic diagrams are not. This gives flexibility in implementing functions. X Y F Z Logic Diagram Logic Equation ZY X F  Truth Table X Y Z Z Y X F   

Logic Circuit Analysis Boolean Algebra and Logic Gates 21 a b c f F = a + b + c b c a f

Logic Circuit Analysis Boolean Algebra and Logic Gates 22 a b c f F = abc b c a f

Logic Circuit Analysis Boolean Algebra and Logic Gates 23 a b c f F = ab + c

Logic Circuit Analysis Boolean Algebra and Logic Gates 24 a c f F = a + c

Logic Circuit Analysis Boolean Algebra and Logic Gates 25 a c F = (a + c). (b+d) f b d

Truth Table (Last Example) Boolean Algebra and Logic Gates 26

Boolean Algebra and Logic Gates 27 F=(a + c). (b+d) b+da + cdcba