The document discusses different types of reasoning including inductive reasoning and deductive reasoning. It provides examples of each type of reasoning and how they are used. Inductive reasoning involves making generalizations based on specific examples, while deductive reasoning uses general statements to logically prove a specific conclusion. The document explains how to use Venn diagrams to determine if a deductive argument is valid or invalid.
Problem solving is the process of finding solutions to difficult or complex issues ,w hile reasoning is the action of thinking about something in a logical, sensible way.
Slides for a lecture by Todd Davies on "Probability", prepared as background material for the Minds and Machines course (SYMSYS 1/PSYCH 35/LINGUIST 35/PHIL 99) at Stanford University. From a video recorded July 30, 2019, as part of a series of lectures funded by a Vice Provost for Teaching and Learning Innovation and Implementation Grant to the Symbolic Systems Program at Stanford, with post-production work by Eva Wallack. Topics include Basic Probability Theory, Conditional Probability, Independence, Philosophical Foundations, Subjective Probability Elicitation, and Heuristics and Biases in Human Probability Judgment.
LECTURE VIDEO: https://youtu.be/tqLluc36oD8
EDITED AND ENHANCED TRANSCRIPT: https://ssrn.com/abstract=3649241
a sample of the 4th Grade Math Book by Jessica Corriere and Robert Richards
The best 4th grade study guide to prepare your student for mathematic exams. The book teaches children to understand basic math concepts, skills, and strategies of the California Common Core Curriculum Standards with detailed step by step explanations to solving typical exam problems. It's like studying with your own private tutor! This book features a user friendly format perfect for browsing, research, and review. Three practice test and answer keys included; covering review topics: Number Sense, Algebra, Geometry, Measurement, Probability and Statistics. All content aligned to state and national standards.
Use inductive reasoning to identify patterns and make conjectures.
Find counterexamples to disprove conjectures.
Understand the difference between inductive and deductive reasoning.
The student is able to (I can):
Use inductive reasoning to identify patterns and make conjectures
Find counterexamples to disprove conjectures
Identify, write, and analyze the truth value of conditional statements.
Write the inverse, converse, and contrapositive of a conditional statement.
Problem solving is the process of finding solutions to difficult or complex issues ,w hile reasoning is the action of thinking about something in a logical, sensible way.
Slides for a lecture by Todd Davies on "Probability", prepared as background material for the Minds and Machines course (SYMSYS 1/PSYCH 35/LINGUIST 35/PHIL 99) at Stanford University. From a video recorded July 30, 2019, as part of a series of lectures funded by a Vice Provost for Teaching and Learning Innovation and Implementation Grant to the Symbolic Systems Program at Stanford, with post-production work by Eva Wallack. Topics include Basic Probability Theory, Conditional Probability, Independence, Philosophical Foundations, Subjective Probability Elicitation, and Heuristics and Biases in Human Probability Judgment.
LECTURE VIDEO: https://youtu.be/tqLluc36oD8
EDITED AND ENHANCED TRANSCRIPT: https://ssrn.com/abstract=3649241
a sample of the 4th Grade Math Book by Jessica Corriere and Robert Richards
The best 4th grade study guide to prepare your student for mathematic exams. The book teaches children to understand basic math concepts, skills, and strategies of the California Common Core Curriculum Standards with detailed step by step explanations to solving typical exam problems. It's like studying with your own private tutor! This book features a user friendly format perfect for browsing, research, and review. Three practice test and answer keys included; covering review topics: Number Sense, Algebra, Geometry, Measurement, Probability and Statistics. All content aligned to state and national standards.
Use inductive reasoning to identify patterns and make conjectures.
Find counterexamples to disprove conjectures.
Understand the difference between inductive and deductive reasoning.
The student is able to (I can):
Use inductive reasoning to identify patterns and make conjectures
Find counterexamples to disprove conjectures
Identify, write, and analyze the truth value of conditional statements.
Write the inverse, converse, and contrapositive of a conditional statement.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
2. Reasoning
Human know to use the Reasoning for supporting
faith or for find the truth. The Mathematical Reasoning are
two important ways .
Inductive Reasoning
Deductive Reasoning
3. Inductive Reasoning
Inductive reasoning is the process of
arriving at a general conclusion based
on observations of specific examples.
Example
You purchased textbooks for 4 classes.
Each book cost more than $50.00.
Conclusion: All college textbooks cost
more than $50.00.
Specific General
4. Ex. 1: Describing a Visual Pattern
Sketch the next figure in the pattern.
1 2 3 4 5
5. Ex. 1: Describing a Visual Pattern -
Solution
The sixth figure in the pattern has 6 squares
in the bottom row.
5 6
6. Ex. 2: Describing a Number Pattern
Describe a pattern in
the sequence of
numbers. Predict the
next number.
1
4
16
64
a = ?
How do you get to
the next number?
That’s right.
Each number is 4 times
the previous number.
So, the next number is
a = 256
7. Ex. 3: Describing a Number Pattern
Describe a pattern in
the sequence of
numbers. Predict the
next number.
-5
-2
4
13
b = ?
How do you get to
the next number?
That’s right.
You add 3 to get to the
next number, then 6,
then 9. To find the fifth
number, you add another
multiple of 3 which is +12
or
b = 25
8. Using Inductive Reasoning
Much of the reasoning you need in
geometry consists of 3 stages:
1. Look for a Pattern: Look at several examples.
Use diagrams and tables to help discover
a pattern.
2. Make a Conjecture. Use the example to
make a general conjecture. Okay, what is
that?
9. Using Inductive Reasoning
A conjecture is an unproven statement that
is based on observations. Discuss the
conjecture with others. Modify the
conjecture, if necessary.
3. Verify the conjecture. Use logical reasoning
to verify the conjecture is true IN ALL CASES.
10. Ex. 4: Making a Conjecture
First odd positive integer:
1 = 12
1 + 3 = 4 = 22
1 + 3 + 5 = 9 = 32
1 + 3 + 5 + 7 = 16 = 42
The sum of the first n odd positive integers is n2.
11. 9 x 1 = 9 and 0 + 9 = 9
9 x 2 = 18 and 1 + 8 = 9
9 x 3 = 27 and 2 + 7 = 9
9 x 4 = 36 and 3 + 6 = 9
and
and
and
and
and
and
9 x 5 = 45 4 + 5 = 9
9 x 6 = 54 5 + 4 = 9
9 x 7 = 63 6 + 3 = 9
9 x 8 = 72 7 + 2 = 9
9 x 9 = 81 8 + 1 = 9
9 x 10 = 90 9 + 0 = 9
16. Note:
To prove that a conjecture is true, you need to
prove it is true in all cases. To prove that a
conjecture is false, you need to provide a
single counter example. A counterexample is
an example that shows a conjecture is false.
17. Ex. 5: Finding a counterexample
Show the conjecture is false by finding a
counterexample.
Conjecture: For all real numbers x, the
expressions x2 is greater than or equal to x.
18. Ex. 5: Finding a counterexample-
Solution
Conjecture: For all real numbers x, the expressions x2
is greater than or equal to x.
The conjecture is false. Here is a counterexample:
(0.5)2 = 0.25, and 0.25 is NOT greater than or equal
to 0.5. In fact, any number between 0 and 1 is a
counterexample.
19. Note:
Not every conjecture is known to be true or
false. Conjectures that are not known to be
true or false are called unproven or
undecided.
20. Ex. 6: Examining an Unproven
Conjecture
In the early 1700’s, a Prussian mathematician names
Goldbach noticed that many even numbers greater
than 2 can be written as the sum of two primes.
Specific cases:
4 = 2 + 2 10 = 3 + 7 16 = 3 + 13
6 = 3 + 3 12 = 5 + 7 18 = 5 + 13
8 = 3 + 5 14 = 3 + 11 20 = 3 + 17
21. Ex. 6: Examining an Unproven
Conjecture
Conjecture: Every even number greater than 2 can
be written as the sum of two primes.
This is called Goldbach’s Conjecture. No one has
ever proven this conjecture is true or found a
counterexample to show that it is false. As of the
writing of this text, it is unknown if this conjecture is
true or false. It is known; however, that all even
numbers up to 4 x 1014 confirm Goldbach’s
Conjecture.
22. Ex. 7: Using Inductive Reasoning in
Real-Life
Moon cycles. A full moon occurs when the
moon is on the opposite side of Earth from
the sun. During a full moon, the moon
appears as a complete circle.
23. Ex. 7: Using Inductive Reasoning in
Real-Life
• Use inductive reasoning and the information
below to make a conjecture about how often
a full moon occurs.
• Specific cases: In 2005, the first six full moons
occur on January 25, February 24, March 25,
April 24, May 23 and June 22.
24. Ex. 7: Using Inductive Reasoning in Real-
Life - Solution
• A full moon occurs every 29 or 30 days.
• This conjecture is true. The moon revolves around
the Earth approximately every 29.5 days.
• Inductive reasoning is very important to the study of
mathematics. You look for a pattern in specific cases
and then you write a conjecture that you think
describes the general case. Remember, though, that
just because something is true for several specific
cases does not prove that it is true in general.
25. Deductive Reasoning
Deductive reasoning is the process of proving
a specific conclusion from one or more
general statements.
A conclusion that is proved true by deductive
reasoning is called a theorem.
ConclusionPremise
Logical Argument
Calculation
Proof
26. Deductive Reasoning
Example:
The catalog states that all entering
freshmen must take a mathematics
placement test.
Conclusion: You will have to take a
mathematics placement test.
You are an entering freshman.
28. Deductive Reasoning
The previous example was wrong
because I never said what I'd do if it
wasn't raining.
– That's what makes deductive reasoning
so difficult; it's easy to get fooled.
– We'll learn more about this when we
study logic.
29. Deductive vs. Inductive Reasoning
The difference between these two types
of reasoning is that
–inductive reasoning tries to generalize
something from some given information,
–deductive reasoning is following a train
of thought that leads to a conclusion.
30. This is called a Venn Diagram for a statement. In the diagram the outside box or
rectangle is used to represent everything. Circles are used to represent general
collections of things. Dots are used to represent specific items in a collection.
There are 3 basic ways that categories of things can fit together.
women
democrats
women democrats
women democrats
All democrats are women. Some democrats are women. No democrats are women.
One circle inside another. Circles overlap. Circles do not touch.
democrats
Laura Bush
women
George
Bush
Laura Bush is a democrat. George Bush is not a woman.
What statement do each of the following Venn Diagrams depict?
31. Putting more than one statement in a diagram. (A previous example)
Hypothesis:
Dr. Daquila voted in the last election.
Only people over 18 years old vote.
Conclusion:
Dr. Daquila is over 18 years old.
Dr.
Daquila
voters
people over 18
IF__THEN__ Statement Construction
Many categorical statements can be made using an if then sentence construction. For example if we
have the statement: All tigers are cats.
cats
tigers
This can be written as an If_then_ statement in
the following way:
If it is a tiger then it is a cat.
This is consistent with how we have been thinking of logical statements. In fact the parts of this
statement even have the same names:
If it is a tiger then it is a cat.
The phrase “it is a tiger” is called the hypothesis. The phrase “it is a cat” is called the conclusion.
32. Can the following statement be deduced?
Hypothesis:
If you are cool then you sit in the back.
If you sit in the back then you can’t see.
Conclusion
If you are cool then you can’t see.
There is only one way this can be drawn! So it can be deduced and the argument is valid!
people who can’t see
people who sit in back
cool people
33. One method that can be used to determine if a statement can be deduced from a collection of
statements that form a hypothesis we draw the Venn Diagram in all possible ways that the hypothesis
would allow. If any of the ways we have drawn is inconsistent with the conclusion the statement can
not be deduced.
Example
Hypothesis:
All football players are talented people.
Pittsburg Steelers are talented people.
Conclusion:
Pittsburg Steelers are football players. (CAN NOT BE DEDUCED and Invalid!)
talented people
football
players
Pittsburg
Steelers
talented people
football players
Pittsburg Steelers
talented people
football
players
Pittsburg
Steelers
Even though one of the ways is consistent with the conclusion there is at least one that is not so this
statement can not be deduced and is not valid.
34. Example
Construct a Venn Diagram to determine the validity of
the given argument.
1. Hypothesis:
All smiling cats talk.
The Cheshire Cat smiles.
Conclusion:
the Cheshire Cat talks.
VALID OR INVALID???
36. Example
Construct a Venn Diagram to determine the validity of
the given argument.
2. Hypothesis:
No one who can afford health insurance is unemployed.
All politicians can afford health insurance.
Conclusion:
no politician is unemployed.
VALID OR INVALID???
37. Examples
People who can afford
Health Care.
Politicians
X
Unemployed
X=politician. The argument is valid.