The document discusses compound propositions and logical expressions. It defines atomic and compound expressions, and explains that compound expressions contain logical connectives. Precedence rules for logical operators are provided. Truth tables are used to evaluate expressions and determine if they are tautologies or contradictions. Examples demonstrate logical equivalences using truth tables, such as De Morgan's laws, distributivity, and contrapositives.
With vocabulary
1. The Statements, Open Sentences, and Trurth Values
2. Negation
3. Compound Statement
4. Equivalence, Tautology, Contradiction, and Contingency
5. Converse, Invers, and Contraposition
6. Making Conclusion
L4-IntroducClick to edit Master title styletion to logic.pptxFahmiOlayah
اسماعيل هنية: نحيClick to edit Master title styleي من ساندنا في #لبنان والعراق واليمن وكل من خرج في العالم أجمع نصرة لنا ولحقوقنا، ونحيي دولة جنوب إفريقيا على رفعها دعوى أمام محكمة العدل الدولية ضد جرائم الاحتلال بحق شعبنا
#غزه_تقاوم
Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities.
With vocabulary
1. The Statements, Open Sentences, and Trurth Values
2. Negation
3. Compound Statement
4. Equivalence, Tautology, Contradiction, and Contingency
5. Converse, Invers, and Contraposition
6. Making Conclusion
L4-IntroducClick to edit Master title styletion to logic.pptxFahmiOlayah
اسماعيل هنية: نحيClick to edit Master title styleي من ساندنا في #لبنان والعراق واليمن وكل من خرج في العالم أجمع نصرة لنا ولحقوقنا، ونحيي دولة جنوب إفريقيا على رفعها دعوى أمام محكمة العدل الدولية ضد جرائم الاحتلال بحق شعبنا
#غزه_تقاوم
Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities.
Opendatabay - Open Data Marketplace.pptxOpendatabay
Opendatabay.com unlocks the power of data for everyone. Open Data Marketplace fosters a collaborative hub for data enthusiasts to explore, share, and contribute to a vast collection of datasets.
First ever open hub for data enthusiasts to collaborate and innovate. A platform to explore, share, and contribute to a vast collection of datasets. Through robust quality control and innovative technologies like blockchain verification, opendatabay ensures the authenticity and reliability of datasets, empowering users to make data-driven decisions with confidence. Leverage cutting-edge AI technologies to enhance the data exploration, analysis, and discovery experience.
From intelligent search and recommendations to automated data productisation and quotation, Opendatabay AI-driven features streamline the data workflow. Finding the data you need shouldn't be a complex. Opendatabay simplifies the data acquisition process with an intuitive interface and robust search tools. Effortlessly explore, discover, and access the data you need, allowing you to focus on extracting valuable insights. Opendatabay breaks new ground with a dedicated, AI-generated, synthetic datasets.
Leverage these privacy-preserving datasets for training and testing AI models without compromising sensitive information. Opendatabay prioritizes transparency by providing detailed metadata, provenance information, and usage guidelines for each dataset, ensuring users have a comprehensive understanding of the data they're working with. By leveraging a powerful combination of distributed ledger technology and rigorous third-party audits Opendatabay ensures the authenticity and reliability of every dataset. Security is at the core of Opendatabay. Marketplace implements stringent security measures, including encryption, access controls, and regular vulnerability assessments, to safeguard your data and protect your privacy.
Chatty Kathy - UNC Bootcamp Final Project Presentation - Final Version - 5.23...John Andrews
SlideShare Description for "Chatty Kathy - UNC Bootcamp Final Project Presentation"
Title: Chatty Kathy: Enhancing Physical Activity Among Older Adults
Description:
Discover how Chatty Kathy, an innovative project developed at the UNC Bootcamp, aims to tackle the challenge of low physical activity among older adults. Our AI-driven solution uses peer interaction to boost and sustain exercise levels, significantly improving health outcomes. This presentation covers our problem statement, the rationale behind Chatty Kathy, synthetic data and persona creation, model performance metrics, a visual demonstration of the project, and potential future developments. Join us for an insightful Q&A session to explore the potential of this groundbreaking project.
Project Team: Jay Requarth, Jana Avery, John Andrews, Dr. Dick Davis II, Nee Buntoum, Nam Yeongjin & Mat Nicholas
Adjusting primitives for graph : SHORT REPORT / NOTESSubhajit Sahu
Graph algorithms, like PageRank Compressed Sparse Row (CSR) is an adjacency-list based graph representation that is
Multiply with different modes (map)
1. Performance of sequential execution based vs OpenMP based vector multiply.
2. Comparing various launch configs for CUDA based vector multiply.
Sum with different storage types (reduce)
1. Performance of vector element sum using float vs bfloat16 as the storage type.
Sum with different modes (reduce)
1. Performance of sequential execution based vs OpenMP based vector element sum.
2. Performance of memcpy vs in-place based CUDA based vector element sum.
3. Comparing various launch configs for CUDA based vector element sum (memcpy).
4. Comparing various launch configs for CUDA based vector element sum (in-place).
Sum with in-place strategies of CUDA mode (reduce)
1. Comparing various launch configs for CUDA based vector element sum (in-place).
Levelwise PageRank with Loop-Based Dead End Handling Strategy : SHORT REPORT ...Subhajit Sahu
Abstract — Levelwise PageRank is an alternative method of PageRank computation which decomposes the input graph into a directed acyclic block-graph of strongly connected components, and processes them in topological order, one level at a time. This enables calculation for ranks in a distributed fashion without per-iteration communication, unlike the standard method where all vertices are processed in each iteration. It however comes with a precondition of the absence of dead ends in the input graph. Here, the native non-distributed performance of Levelwise PageRank was compared against Monolithic PageRank on a CPU as well as a GPU. To ensure a fair comparison, Monolithic PageRank was also performed on a graph where vertices were split by components. Results indicate that Levelwise PageRank is about as fast as Monolithic PageRank on the CPU, but quite a bit slower on the GPU. Slowdown on the GPU is likely caused by a large submission of small workloads, and expected to be non-issue when the computation is performed on massive graphs.
Data Centers - Striving Within A Narrow Range - Research Report - MCG - May 2...pchutichetpong
M Capital Group (“MCG”) expects to see demand and the changing evolution of supply, facilitated through institutional investment rotation out of offices and into work from home (“WFH”), while the ever-expanding need for data storage as global internet usage expands, with experts predicting 5.3 billion users by 2023. These market factors will be underpinned by technological changes, such as progressing cloud services and edge sites, allowing the industry to see strong expected annual growth of 13% over the next 4 years.
Whilst competitive headwinds remain, represented through the recent second bankruptcy filing of Sungard, which blames “COVID-19 and other macroeconomic trends including delayed customer spending decisions, insourcing and reductions in IT spending, energy inflation and reduction in demand for certain services”, the industry has seen key adjustments, where MCG believes that engineering cost management and technological innovation will be paramount to success.
MCG reports that the more favorable market conditions expected over the next few years, helped by the winding down of pandemic restrictions and a hybrid working environment will be driving market momentum forward. The continuous injection of capital by alternative investment firms, as well as the growing infrastructural investment from cloud service providers and social media companies, whose revenues are expected to grow over 3.6x larger by value in 2026, will likely help propel center provision and innovation. These factors paint a promising picture for the industry players that offset rising input costs and adapt to new technologies.
According to M Capital Group: “Specifically, the long-term cost-saving opportunities available from the rise of remote managing will likely aid value growth for the industry. Through margin optimization and further availability of capital for reinvestment, strong players will maintain their competitive foothold, while weaker players exit the market to balance supply and demand.”
2. LOGICAL EXPRESSIONS
Any proposition must somehow be expressed
verbally, graphically, or by a string of characters.
A proposition expressed by a string of characters is
called a logical expression or a formula.
It can either be atomic or compound.
Atomic expression is consisting of a single
propositional variable and it represents atomic
proposition.
Compound expressions contain at least one
connective, and they represent compound
propositions.
3. All expressions containing identifiers that
represent expressions are called schemas.
If A = (P ^ Q) and if B = (P v Q), then the schema
(A => B) stands for ((P ^ Q) => (P v Q)).
4. PRECEDENCE RULES
1. ~
2. ^
3. v
4. =>
5. <=>
~P v Q is equal to (~P) v Q not ~(P v Q)
P ^ Q v R is equal to (P ^ Q) v R
P => Q v R is equal to P=> (Q v R)
M <=> F => C is equal to M <=> (F => C)
P => Q => R is equal to (P => Q) => R because all
binary logical connectives are left associative.
5. Prefix –precedes it operand (start)
Infix –inserted between the operands (middle)
Postfix –follows its operands (end)
Examples of Infix, Prefix, and Postfix
Infix Expression Prefix Expression Postfix Expression
a + b , p ^ q + a b, ~p a b +
a + b * c , p ^ q v r + a * b c, ~q a b c * +
6. EVALUATION OF EXPRESSIONS
AND TRUTH TABLES
“If you take a class in computer, and if you do not
understand logic, you will not pass.”
P: You take a class in computer.
Q: You understand logic.
R: You pass.
We want to know exactly when this statement is
true or false. To do this we use the symbols P, Q,
and R. Using this, the statement in question
becomes (P ^ ~Q) => ~R.
7. TRUTH TABLE FOR (P ^ ~Q) => ~R:
P Q R ~Q P ^ ~Q ~R (P ^ ~Q)
=> ~R
T T T F F F T
T T F F F T T
T F T T T F F
T F F T T T T
F T T F F F T
F T F F F T T
F F T T F F T
F F F T F T T
8. SEATWORK
Given P and Q are true and R and S are false,
find the truth-values of the ff expressions:
1. P v (Q ^ R)
2. (P ^ (Q ^ R)) v ~((P v Q) ^ (R v S))
9. SOLUTION
1. P v (Q ^ R)
2. (P ^ (Q ^ R)) v ~((P v Q) ^ (R v S))
P Q R Q^R P v (Q ^ R)
T T F F T
T T F F T
T T F F T
T T F F T
P Q R S (Q ^
R)
(P ^ (Q ^
R))
(P v
Q)
(R v
S)
((P v Q) ^
(R v S))
~((P v Q) ^
(R v S))
v
T T F F F F T F F T T
T T F F F F T F F T T
T T F F F F T F F T T
T T F F F F T F F T T
11. Tautology gives all answers with truth value of
TRUE.
Contradiction gives all answers with truth
value of FALSE.
Contingency - proposition that is neither a
tautology nor a contradiction
12. Tautology and Logical equivalence
Definitions:
A compound proposition that is always True is
called a tautology.
Two propositions p and q are logically equivalent
if their truth tables are the same.
Namely, p and q are logically equivalent if p ↔ q
is a tautology. If p and q are logically equivalent,
we write p ≡ q.
13. EXAMPLES OF LOGICAL EQUIVALENCE
Example: Look at the following two compound
propositions: p → q and q ∨ ¬p.
The last column of the two truth tables are
identical. Therefore (p → q) and (q ∨ ¬p) are
logically equivalent.
So (p → q) ↔ (q ∨ ¬p) is a tautology.
Thus: (p → q)≡ (q ∨ ¬p).
Example:
By using truth table, prove p ⊕ q ≡ ¬ (p ↔ q).
14. DE MORGAN LAW
We have a number of rules for logical
equivalence.
For example: De Morgan Law:
¬(p ∧ q) ≡ ¬p ∨ ¬q (1)
¬(p ∨ q) ≡ ¬p ∧ ¬q (2)
The following is the truth table proof for (1). The
proof for (2) is similar
15. DISTRIBUTIVITY
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) (1)
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) (2)
The following is the truth table proof of (1). The
proof of (2) is similar.
16. CONTRAPOSITIVES
The proposition ¬q → ¬p is called the
Contrapositive of the proposition p → q. They
are logically equivalent.
p → q ≡ ¬q → ¬p
