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Chapter 2 - Part 1 - PPT - Mano & Kime - 2nd EdSlide 1 Lecture 4 Binary Logic and Logic GatesLecture 4 Binary Logic and Logic Gates
Slide 2 OutlineOutline Binary Logic and Variables Logical Operations Truth Tables Logic Implementation Logic Gates
Slide 3 Binary Logic and GatesBinary 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 foundation for designing and analyzing digital systems!
Slide 4 Binary VariablesBinary 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 X1 for now RESET, START_IT, or ADD1 later
Slide 5 Logical OperationsLogical 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 overbar ( ¯ ), a single quote mark (') after, or (~) before the variable.
Slide 6 Notation ExamplesExamples: 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: 1 + 1 = 2 (read one plus one equals two ) is not the same as 1 + 1 = 1 (read 1 or 1 equals 1 ). = B A Y × y x z + = A X =
Slide 7 Operator DefinitionsOperator 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 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1 NOT 1 0 = 0 1 =
Slide 8 Truth Tables0 1 1 0 X NOT X Z = Truth Tables Truth table - a 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: 1 1 1 0 0 1 0 1 0 0 0 0 Z = X·Y Y X AND
Slide 9 Logic Function ImplementationUsing Switches For inputs: logic 1 is switch closed logic 0 is switch open For outputs: logic 1 is light on logic 0 is light off. NOT uses a switch such that: 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
Slide 10 Logic Function Implementation (Continued)Example: Logic Using Switches Light is on (L = 1) for L(A, B, C, D) = and off (L = 0), otherwise. Useful model for relay circuits and for CMOS gate circuits, the foundation of current digital logic technology Logic Function Implementation (Continued) B A D C A ((B C') + D) = A B C' + A D
L (A, B, C, D) = A ((B C') + D) = A B C' + A D
Slide 11 Logic GatesLogic 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.
Slide 12 Logic Gates (continued)Logic Gates (continued) Implementation of logic gates with transistors (See Reading Supplement - CMOS Circuits) Transistor or tube implementations of logic functions are called logic gates or just gates Transistor gate circuits can be modeled by switch circuits Adobe Systems
The transistor without the bubble on its input is an N-type field effect transistor. It acts like a closed switch between its top and bottom terminals with an H (1) applied to its input on its left. It acts like an open switch with an L (0) applied to its input. The transistor with the bubble on its input is a P-type field effect transistor. The +V at the top provides an H (1) and the Ground symbol at the bottom provides an L (0). By modeling the two types of field effect transistors as switches, one can see how the series and parallel interconnections can produce 1 s and 0 s on the outputs on the right in response to applied 1 s and 0 s on the inputs on the left. NOR and NAND are OR and AND, each followed by a NOT respectively.
Slide 13 Logic Gate Symbols and BehaviorAdobe Systems Logic Gate Symbols and Behavior Logic gates have special symbols: And waveform behavior in time as follows:
Slide 14 Logic Diagrams and ExpressionsLogic Diagrams and Expressions Boolean equations, truth tables and logic diagrams describe the same function! Truth tables are unique; expressions and logic diagrams are not. This gives flexibility in implementing functions. X Y F Z Logic Diagram Equation Z Y X F + = Truth Table 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 X Y Z Z Y X F × + =
Slide 15 SummarySummary Binary Logic and Variables Logical Operations Truth Tables Logic Implementation Logic Gates
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