Logical implication - Necessary and Sufficient conditions
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Discrete Mathematical Structures Fundamentals of Logic Logical implication - Necessary and Sufficient conditions
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Logic&proof
This document summarizes Chapter 1 of the textbook "Discrete Mathematics" by R. Johnsonbaugh. It covers the topics of logic, proofs, and propositional logic. Key points include:
- Logic is the study of correct reasoning and is used in mathematics and computer science.
- A proposition is a statement that can be determined as true or false. Connectives like AND, OR, and NOT can combine propositions.
- Truth tables define the truth values of compound propositions formed from connectives.
- Quantifiers like "for all" and "there exists" are used to make universal and existential statements.
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The document discusses different types of logical conditionals:
- A conditional statement relates two propositions using "if...then..."
- The converse flips the order of the propositions in the conditional
- The inverse negates both propositions
- The contrapositive applies both converse and inverse operations
Several examples are provided to illustrate each type of conditional. Conditionals, their converses, inverses, and contrapositives can be represented using geometric shapes like triangles and polygons.
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The order of the given matrix is 2×3. So the maximum no. of elements is 2×3 = 6.
The correct option is B.
The element a32 belongs to 3rd row and 2nd column.
The correct option is B.
3. A matrix whose each diagonal element is unity and all other elements are zero is called
A) Identity matrix B) Unit matrix C) Scalar matrix D) Diagonal matrix
4. A matrix whose each row sums to unity is called
A) Row matrix B) Column matrix C) Unit matrix D) Stochastic matrix
5. The sum of all the elements on the principal diagonal of a square
Logic (PROPOSITIONS)
This document discusses propositional logic and truth tables. It defines primitive and compound propositions. Logical connectives like negation, disjunction, and conjunction are explained. Propositional variables are used to represent statements that can be true or false. Truth tables list all possible combinations of true and false values for propositional variables and determine the truth value of compound statements formed from logical connectives. The number of rows in a truth table is determined by 2 to the power of the number of propositional variables. Several examples of truth tables are given for logical connectives like negation, disjunction, conjunction, implication, and biconditional.
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This document introduces basic concepts in propositional logic, including:
1. Propositions are declarative statements that are either true or false. Compound propositions consist of simple propositions connected by logical operators like AND and OR.
2. Truth tables define logical connectives like conjunction, disjunction, conditional, biconditional, and negation. Equivalences between statements can be shown through truth tables.
3. Logical implications can be proven without truth tables by showing that if the antecedent is true, the consequent must also be true. Dual statements and De Morgan's laws are also introduced.
Basic Connectives and Truth Tables.ppt
This document discusses logic and truth tables which are used in mathematics and computer science. It defines primitive statements, logical connectives like conjunction, disjunction, negation, implication and biconditional. Truth tables are used to determine the truth values of compound statements formed using these connectives. Examples are given to show how compound statements can be written symbolically and their truth values determined from truth tables. Decision structures like if-then and if-then-else used in programming languages are also discussed.
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This document summarizes Chapter 1 of the textbook "Discrete Mathematics" by R. Johnsonbaugh. It covers the topics of logic, proofs, and propositional logic. Key points include:
- Logic is the study of correct reasoning and is used in mathematics and computer science.
- A proposition is a statement that can be determined as true or false. Connectives like AND, OR, and NOT can combine propositions.
- Truth tables define the truth values of compound propositions formed from connectives.
- Quantifiers like "for all" and "there exists" are used to make universal and existential statements.
- A proof is a logical argument establishing the truth of a theorem using definitions, ax
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Propositional logic is presented. A proposition is a statement that can be either true or false. Logical connectives like negation, conjunction, disjunction, conditional, biconditional, NOR, NAND and XOR are used to combine propositions. A tautology is a proposition that is always true, while a contradiction is always false. Truth tables are used to determine if a proposition is a tautology, contradiction or contingency. Logical equivalence means that two propositions have the same truth values according to their truth tables.
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1st Test - If then, converse, inverse and contrapositive
The document discusses different types of logical conditionals:
- A conditional statement relates two propositions using "if...then..."
- The converse flips the order of the propositions in the conditional
- The inverse negates both propositions
- The contrapositive applies both converse and inverse operations
Several examples are provided to illustrate each type of conditional. Conditionals, their converses, inverses, and contrapositives can be represented using geometric shapes like triangles and polygons.
Introduction of matrices
The order of the given matrix is 2×3. So the maximum no. of elements is 2×3 = 6.
The correct option is B.
The element a32 belongs to 3rd row and 2nd column.
The correct option is B.
3. A matrix whose each diagonal element is unity and all other elements are zero is called
A) Identity matrix B) Unit matrix C) Scalar matrix D) Diagonal matrix
4. A matrix whose each row sums to unity is called
A) Row matrix B) Column matrix C) Unit matrix D) Stochastic matrix
5. The sum of all the elements on the principal diagonal of a square
Logic (PROPOSITIONS)
This document discusses propositional logic and truth tables. It defines primitive and compound propositions. Logical connectives like negation, disjunction, and conjunction are explained. Propositional variables are used to represent statements that can be true or false. Truth tables list all possible combinations of true and false values for propositional variables and determine the truth value of compound statements formed from logical connectives. The number of rows in a truth table is determined by 2 to the power of the number of propositional variables. Several examples of truth tables are given for logical connectives like negation, disjunction, conjunction, implication, and biconditional.
Logic and proof
This document introduces basic concepts in propositional logic, including:
1. Propositions are declarative statements that are either true or false. Compound propositions consist of simple propositions connected by logical operators like AND and OR.
2. Truth tables define logical connectives like conjunction, disjunction, conditional, biconditional, and negation. Equivalences between statements can be shown through truth tables.
3. Logical implications can be proven without truth tables by showing that if the antecedent is true, the consequent must also be true. Dual statements and De Morgan's laws are also introduced.
Basic Connectives and Truth Tables.ppt
This document discusses logic and truth tables which are used in mathematics and computer science. It defines primitive statements, logical connectives like conjunction, disjunction, negation, implication and biconditional. Truth tables are used to determine the truth values of compound statements formed using these connectives. Examples are given to show how compound statements can be written symbolically and their truth values determined from truth tables. Decision structures like if-then and if-then-else used in programming languages are also discussed.
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Here are the definitions of conditional and biconditional propositions:
Conditional proposition: A proposition of the form "If p, then q" which is symbolized as p → q. It is only false when p is true and q is false. It is true in all other cases.
Biconditional proposition: A proposition that links two statements such that the truth or falsity of one depends on the truth or falsity of the other. It is symbolized as p ↔ q and reads as "p if and only if q". It is true when p and q are either both true or both false, and false otherwise.
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6. Making Conclusion
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The document defines and provides examples of different types of matrices, including:
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- Rectangular matrices, where the number of rows does not equal the number of columns.
- Row matrices, with only one row.
- Column matrices, with only one column.
- Null or zero matrices, with all elements equal to zero.
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Here are the definitions of conditional and biconditional propositions:
Conditional proposition: A proposition of the form "If p, then q" which is symbolized as p → q. It is only false when p is true and q is false. It is true in all other cases.
Biconditional proposition: A proposition that links two statements such that the truth or falsity of one depends on the truth or falsity of the other. It is symbolized as p ↔ q and reads as "p if and only if q". It is true when p and q are either both true or both false, and false otherwise.
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With vocabulary
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2. Negation
3. Compound Statement
4. Equivalence, Tautology, Contradiction, and Contingency
5. Converse, Invers, and Contraposition
6. Making Conclusion
Matrix.
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- Square matrices, where the number of rows equals the number of columns.
- Rectangular matrices, where the number of rows does not equal the number of columns.
- Row matrices, with only one row.
- Column matrices, with only one column.
- Null or zero matrices, with all elements equal to zero.
- Diagonal matrices, with all elements equal to zero except those on the main diagonal.
The document also discusses transpose, adjoint, and addition of matrices.
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Discrete Mathematical Structures - Fundamentals of Logic - Principle of duality
Recently uploaded
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODEL
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, SerbiaACEP Magazine edition 4th launched on 05.06.2024
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECT
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
[JPP-1] - (JEE 3.0) - Kinematics 1D - 14th May..pdf
Kinematics 11th jpp- 01. ( Solved ) unacademy namo kaul on 14th may...
Modelagem de um CSTR com reação endotermica.pdf
Modelagem em função de transferencia. CSTR não-linear.
Recycled Concrete Aggregate in Construction Part III
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
DfMAy 2024 - key insights and contributions
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMS
The smart irrigation system represents an innovative approach to optimize water usage in agricultural and landscaping practices. The integration of cutting-edge technologies, including sensors, actuators, and data analysis, empowers this system to provide accurate monitoring and control of irrigation processes by leveraging real-time environmental conditions. The main objective of a smart irrigation system is to optimize water efficiency, minimize expenses, and foster the adoption of sustainable water management methods. This paper conducts a systematic risk assessment by exploring the key components/assets and their functionalities in the smart irrigation system. The crucial role of sensors in gathering data on soil moisture, weather patterns, and plant well-being is emphasized in this system. These sensors enable intelligent decision-making in irrigation scheduling and water distribution, leading to enhanced water efficiency and sustainable water management practices. Actuators enable automated control of irrigation devices, ensuring precise and targeted water delivery to plants. Additionally, the paper addresses the potential threat and vulnerabilities associated with smart irrigation systems. It discusses limitations of the system, such as power constraints and computational capabilities, and calculates the potential security risks. The paper suggests possible risk treatment methods for effective secure system operation. In conclusion, the paper emphasizes the significant benefits of implementing smart irrigation systems, including improved water conservation, increased crop yield, and reduced environmental impact. Additionally, based on the security analysis conducted, the paper recommends the implementation of countermeasures and security approaches to address vulnerabilities and ensure the integrity and reliability of the system. By incorporating these measures, smart irrigation technology can revolutionize water management practices in agriculture, promoting sustainability, resource efficiency, and safeguarding against potential security threats.
RAT: Retrieval Augmented Thoughts Elicit Context-Aware Reasoning in Long-Hori...
RAT: Retrieval Augmented Thoughts Elicit Context-Aware Reasoning in Long-Horizon Generation
6th International Conference on Machine Learning & Applications (CMLA 2024)
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
IEEE Aerospace and Electronic Systems Society as a Graduate Student Member
IEEE Aerospace and Electronic Systems Society as a Graduate Student Member
Embedded machine learning-based road conditions and driving behavior monitoring
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
Recently uploaded (20)
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODEL
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODEL
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECT
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECT
[JPP-1] - (JEE 3.0) - Kinematics 1D - 14th May..pdf
[JPP-1] - (JEE 3.0) - Kinematics 1D - 14th May..pdf
BPV-GUI-01-Guide-for-ASME-Review-Teams-(General)-10-10-2023.pdf
BPV-GUI-01-Guide-for-ASME-Review-Teams-(General)-10-10-2023.pdf
Recycled Concrete Aggregate in Construction Part III
Recycled Concrete Aggregate in Construction Part III
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMS
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMS
RAT: Retrieval Augmented Thoughts Elicit Context-Aware Reasoning in Long-Hori...
RAT: Retrieval Augmented Thoughts Elicit Context-Aware Reasoning in Long-Hori...
6th International Conference on Machine Learning & Applications (CMLA 2024)
6th International Conference on Machine Learning & Applications (CMLA 2024)
ML Based Model for NIDS MSc Updated Presentation.v2.pptx
ML Based Model for NIDS MSc Updated Presentation.v2.pptx
IEEE Aerospace and Electronic Systems Society as a Graduate Student Member
IEEE Aerospace and Electronic Systems Society as a Graduate Student Member
Generative AI leverages algorithms to create various forms of content
Generative AI leverages algorithms to create various forms of content
Embedded machine learning-based road conditions and driving behavior monitoring
Embedded machine learning-based road conditions and driving behavior monitoring
basic-wireline-operations-course-mahmoud-f-radwan.pdf
basic-wireline-operations-course-mahmoud-f-radwan.pdf
Logical implication - Necessary and Sufficient conditions
- 1. By, Lakshmi R Asst. Professor Dept. of ISE
- 2. Consider p →q.That is, if p, then q then, p is sufficient condition for q and q is necessary condition for p Eg: “If there are animals, then oxygen is present” Presence of oxygen is necessary for animals’ life Presence of animals is sufficient condition to say oxygen is present. Oxygen can be present without the presence of animals. Lakshmi R, Asst. Professor, Dept. of ISE
- 3. When a hypothetical p →q is such that q is true whenever p is true, then we say that “p logically implies q” This is symbolically represented as p ⇒q ‘⇒’ denotes logical implication Consider two propositions p and q whose truth values are interrelated. Then, for p →q to be a logical implication, following should hold good i. p ⇒q ii. p is sufficient for q iii. q is necessary for p Lakshmi R, Asst. Professor, Dept. of ISE
- 4. 1. If a number is divisible by 6, then it is divisible by 3. 2. If a shape is a parallelogram, then its diagonals bisect each other. 3. If a triangle is not isosceles, then it is not equilateral. 4. If a quadrilateral is a square, then it is a rectangle. Lakshmi R, Asst. Professor, Dept. of ISE
- 5. If a number is divisible by 6, then it is divisible by 3. A necessary condition for a number to be divisible by 6 is that it is divisible by 3. A sufficient condition for a number to be divisible by 3 is that number is divisible by 6. Solution: Lakshmi R, Asst. Professor, Dept. of ISE
- 6. If a shape is a parallelogram, then its diagonals bisect each other. A necessary condition for a shape to be parallelogram is that its diagonals bisect each other. A sufficient condition for diagonals of a shape to bisect each other is that it is a parallelogram. Solution: Lakshmi R, Asst. Professor, Dept. of ISE
- 7. Lakshmi R, Asst. Professor, Dept. of ISE If a triangle is not isosceles, then it is not equilateral. Solution: Consider contrapositive of the above statement. If a triangle is equilateral, then it is isosceles A sufficient condition triangle to be isosceles is that it is equilateral. Note: triangle being isosceles does not guarantee that it is equilateral. A necessary condition for a triangle to be equilateral is that it is isosceles.
- 8. Lakshmi R, Asst. Professor, Dept. of ISE If a quadrilateral is a square, then it is a rectangle. Shape Rules Quadrilateral 4 sides Parallelogram 4 sides Opposite sides are parallel Rectangle 4 sides Opposite sides are parallel Equiangular Square 4 sides Opposite sides are parallel Equiangular All sides are equal Solve it!!