Engineers Garage

  • Projects and Tutorials
    • Electronic Projects
      • 8051
      • Arduino
      • ARM
      • AVR
      • PIC
      • Raspberry pi
      • STM32
    • Tutorials
    • Circuit Design
    • Project Videos
    • Components
  • Articles
    • Tech Articles
    • Insight
    • Invention Stories
    • How to
    • What Is
  • News
    • Electronic Products News
    • DIY Reviews
    • Guest Post
  • Forums
    • EDABoard.com
    • Electro-Tech-Online
    • EG Forum Archive
  • Digi-Key Store
    • Cables, Wires
    • Connectors, Interconnect
    • Discrete
    • Electromechanical
    • Embedded Computers
    • Enclosures, Hardware, Office
    • Integrated Circuits (ICs)
    • Isolators
    • LED/Optoelectronics
    • Passive
    • Power, Circuit Protection
    • Programmers
    • RF, Wireless
    • Semiconductors
    • Sensors, Transducers
    • Test Products
    • Tools
  • EE Resources
    • DesignFast
    • LEAP Awards
    • Oscilloscope Product Finder
    • White Papers
    • Webinars
  • EE Learning Center
    • Design Guides
      • WiFi & the IOT Design Guide
      • Microcontrollers Design Guide
      • State of the Art Inductors Design Guide
  • Women in Engineering

All Boolean Logical Operations – DE Part 6

By Hai Prasaath K

In the previous tutorial, various boolean postulates and theorems were discussed. These theorems and postulates are useful in deducing a boolean expression. It was also discussed that for n number of variables, there can be maximum 2^2n boolean functions. So, there can be maximum 16 (2^4) boolean functions between two boolean variables. A boolean variable represents a singular binary data source in digital electronics i.e. a single bit or serial stream of bits. So, there can be maximum 16 logical functions in digital circuits. Let us learn about all the logical operations.

Representational Image of All Boolean Logic Operations

Fig. 1: Representational Image of All Boolean Logic Operations

There can be 16 possible combinations of outcomes between two boolean variables, say X and Y. These outcomes are shown in the following truth table –

Truth Table Listing All Possible Boolean Functions for Two Variables

                                      Fig. 2: Truth Table listing all possible boolean functions for two variables

Corresponding to the following 16 combinations of possible outcomes, there are the following logical operations –

F0: Null – The output being false for all inputs of boolean variables is called Null. This is equivalent to binary constant 0. The function is equivalent to boolean expression F = 0.Corresponding to the following 16 combinations of possible outcomes, there are the following logical operations –

F1: AND – The output being true when both inputs of boolean variables is true otherwise output being false is called AND operation. This is one of the basic logical operations. It is represented by dot (.) operator in the boolean algebra. The function is equivalent to boolean expression F = xy or F = x.y.

F2: Inhibition – The output being true for one variable being true but not the other is called Inhibition. For function F2, the outcome is true provided x is true but not y. In boolean algebraic notation, it is written as x/y. The function is equivalent to boolean expression F = xy’.

F3: Transfer – The output being true if and only if one of the boolean variables is true is called transfer. For function F3, the outcome is true provided only if x is true irrespective of y. In boolean algebraic notation, it is represented simply by writing x. The function is equivalent to boolean expression F = x.

F4: Inhibition – The output being true for one variable being true but not the other is called Inhibition. The function F4 is similar to function F2. For F4, the outcome is true provided y is true but not x. In boolean algebraic notation, it is written as y/x. The function is equivalent to boolean expression F = x’y.

F5: Transfer – The output being true if and only if one of the boolean variables is true is called transfer. The function F5 is similar to function F3. For F5, the outcome is true provided only if y is true irrespective of x. In boolean algebraic notation, it is represented simply by writing y. The function is equivalent to boolean expression F = y.

F6: Exclusive-OR   – The output being true only when either one of the boolean variables is true but not the other is called Exclusive-OR. For this function, the outcome is true provided x or y is true but not both. In boolean algebra, the EX-OR operation is  represented  by  operator. The function is equivalent to boolean expression F = xy‘ + x’y.

F7: OR – The output being true if either or both of the boolean variables is true is called OR operation. This is one of the basic logical operations. It is represented by Plus (+) operator in the boolean algebra. The function is equivalent to boolean expression F = x + y.

F8: NOR – The inverse of OR operation is called NOR. Its outcome is true only when both variables are false. In boolean algebra, it is represented by down arrow  . It is written as . The function is equivalent to boolean expression F = (x + y)’.

F9: Equivalence – The output being true only when both variables are either true or both are false is called equivalence. This is inverse of EX-OR, so is also called Exclusive NOR (EX-NOR). In  boolean algebra, it is represented as (xy)’. The function is equivalent to boolean expression F = xy + x’y‘.

F10: Complement – In function F10, the output is complement of one of the boolean variables. The outcome is true if y is false irrespective of x. In boolean algebra, it is represented as NOT of Y. It is written as y’.  The function is equivalent to boolean expression F = y’.

F11: Implication – The output being true if either one of the boolean variable is false or the other is true is called implication. In Function F11, the output is true if either y is false or x is true. In boolean algebra, it is represented as x  y. The function is equivalent to boolean expression F = x + y’.

F12: Complement – In function F12, the output is complement of one of the boolean variables. The outcome is true if x is false irrespective of y. In boolean algebra, it is represented as NOT of x. It is written as x’. The function is equivalent to boolean expression F = x’.

F13: Implication – The output being true if either one of the boolean variable is false or the other is true is called implication. In Function F13, the output is true if either x is false or y is true. In boolean algebra, it is represented as y  x. The function is equivalent to boolean expression F = x’ + y.

F14: NAND – The inverse of AND operation is called NAND. It is one of the basic logical operations. The output of NAND is false if both boolean variables are true otherwise it is true. In boolean algebra, it is represented by upward arrow ( ). It is written as x  y. The function is equivalent to boolean expression F = (xy)’.

F15: Identity – The output being true for all inputs of boolean variables is called Identity. This is equivalent to binary constant 1. The function is equivalent to boolean expression F = 1.

All the possible logical operations between two boolean variables are summarized in the following table –

Table Listing all Boolean Functions with Two Variables

                                               Fig. 3: Table listing all boolean functions with two variables

So, the basic logical operations – AND, OR, NOT, XOR, XNOR, NAND, NOR along with buffer, inhibition, implication, null and identity operations are the only logical (boolean) operations. The complement and transfer operations are unary operations which work on a single operand.

In the previous tutorial, a boolean expression was minimized usingtruth table. But that is not a standard or systematic way to minimize a boolean expression. In the next tutorial, learn about Gate Level Minimization that includes important mapping techniques for simplification of a boolean expression. The K-Map, VEM and QM techniques are important gate level minimization techniques that must be known for gate level implementation of boolean expressions and so the digital circuits. Once, the gate level minimization techniques are known, practical digital circuits can be designed. So, the basic logical operations – AND, OR, NOT, XOR, XNOR, NAND, NOR along with buffer, inhibition, implication, null and identity operations are the only logical (boolean) operations. The complement and transfer operations are unary operations which work on a single operand.


Filed Under: Featured Contributions, Tutorials

 

Questions related to this article?
👉Ask and discuss on Electro-Tech-Online.com and EDAboard.com forums.



Tell Us What You Think!! Cancel reply

You must be logged in to post a comment.

HAVE A QUESTION?

Have a technical question about an article or other engineering questions? Check out our engineering forums EDABoard.com and Electro-Tech-Online.com where you can get those questions asked and answered by your peers!


Featured Tutorials

  • Introduction to Brain Waves & its Types (Part 1/13)
  • Understanding NeuroSky EEG Chip in Detail (Part 2/13)
  • Performing Experiments with Brainwaves (Part 3/13)
  • Amplification of EEG Signal and Interfacing with Arduino (Part 4/13)
  • Controlling Led brightness using Meditation and attention level (Part 5/13)
  • Control Motor’s Speed using Meditation and Attention Level of Brain (Part 6/13)

Stay Up To Date

Newsletter Signup

Sign up and receive our weekly newsletter for latest Tech articles, Electronics Projects, Tutorial series and other insightful tech content.

EE Training Center Classrooms

EE Classrooms

Recent Articles

  • What are the battery-selection criteria for low-power design?
  • Key factors to optimize power consumption in an embedded device
  • EdgeLock A5000 Secure Authenticator
  • How to interface a DS18B20 temperature sensor with MicroPython’s Onewire driver
  • Introduction to Brain Waves & its Types (Part 1/13)

Most Popular

5G 555 timer circuit 8051 ai Arduino atmega16 automotive avr bluetooth dc motor display Electronic Part Electronic Parts Fujitsu ic infineontechnologies integratedcircuit Intel IoT ir lcd led maximintegratedproducts microchip microchiptechnology Microchip Technology microcontroller microcontrollers mosfet motor powermanagement Raspberry Pi remote renesaselectronics renesaselectronicscorporation Research samsung semiconductor sensor software STMicroelectronics switch Technology vishayintertechnology wireless

RSS EDABOARD.com Discussions

  • Vco cadencd
  • Thyristor Gate Drive
  • Variable Phase shift control circuit for PWM circuit
  • Passive Harmonics Filter
  • Constraining a combo path (Synopsys DC)

RSS Electro-Tech-Online.com Discussions

  • HV Diodes
  • Disabled son needs advice please
  • Question about ultrasonic mist maker
  • RF modules which can handle high number of bytes per second
  • DIY bluetooth speaker
Engineers Garage
  • Analog IC TIps
  • Connector Tips
  • DesignFast
  • EDABoard Forums
  • EE World Online
  • Electro-Tech-Online Forums
  • Microcontroller Tips
  • Power Electronic Tips
  • Sensor Tips
  • Test and Measurement Tips
  • 5G Technology World
  • About Us
  • Contact Us
  • Advertise

Copyright © 2022 WTWH Media LLC. All Rights Reserved. The material on this site may not be reproduced, distributed, transmitted, cached or otherwise used, except with the prior written permission of WTWH Media
Privacy Policy | Advertising | About Us

Search Engineers Garage

  • Projects and Tutorials
    • Electronic Projects
      • 8051
      • Arduino
      • ARM
      • AVR
      • PIC
      • Raspberry pi
      • STM32
    • Tutorials
    • Circuit Design
    • Project Videos
    • Components
  • Articles
    • Tech Articles
    • Insight
    • Invention Stories
    • How to
    • What Is
  • News
    • Electronic Products News
    • DIY Reviews
    • Guest Post
  • Forums
    • EDABoard.com
    • Electro-Tech-Online
    • EG Forum Archive
  • Digi-Key Store
    • Cables, Wires
    • Connectors, Interconnect
    • Discrete
    • Electromechanical
    • Embedded Computers
    • Enclosures, Hardware, Office
    • Integrated Circuits (ICs)
    • Isolators
    • LED/Optoelectronics
    • Passive
    • Power, Circuit Protection
    • Programmers
    • RF, Wireless
    • Semiconductors
    • Sensors, Transducers
    • Test Products
    • Tools
  • EE Resources
    • DesignFast
    • LEAP Awards
    • Oscilloscope Product Finder
    • White Papers
    • Webinars
  • EE Learning Center
    • Design Guides
      • WiFi & the IOT Design Guide
      • Microcontrollers Design Guide
      • State of the Art Inductors Design Guide
  • Women in Engineering