This document discusses inductive and deductive reasoning. It provides examples of each type of reasoning and how they differ. Inductive reasoning involves examining specific examples to reach a general conclusion, while deductive reasoning starts with a general statement and uses logical principles to reach a specific conclusion. The document also provides examples of logic puzzles and how deductive reasoning can be used to solve them by organizing the clues and data provided.
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.
3 PROBABILITY TOPICSFigure 3.1 Meteor showers are rare, .docxtamicawaysmith
3 | PROBABILITY TOPICS
Figure 3.1 Meteor showers are rare, but the probability of them occurring can be calculated. (credit: Navicore/flickr)
Introduction
Chapter Objectives
By the end of this chapter, the student should be able to:
• Understand and use the terminology of probability.
• Determine whether two events are mutually exclusive and whether two events are independent.
• Calculate probabilities using the Addition Rules and Multiplication Rules.
• Construct and interpret Contingency Tables.
• Construct and interpret Venn Diagrams.
• Construct and interpret Tree Diagrams.
It is often necessary to "guess" about the outcome of an event in order to make a decision. Politicians study polls to guess
their likelihood of winning an election. Teachers choose a particular course of study based on what they think students can
comprehend. Doctors choose the treatments needed for various diseases based on their assessment of likely results. You
may have visited a casino where people play games chosen because of the belief that the likelihood of winning is good. You
may have chosen your course of study based on the probable availability of jobs.
You have, more than likely, used probability. In fact, you probably have an intuitive sense of probability. Probability deals
with the chance of an event occurring. Whenever you weigh the odds of whether or not to do your homework or to study
for an exam, you are using probability. In this chapter, you will learn how to solve probability problems using a systematic
approach.
Your instructor will survey your class. Count the number of students in the class today.
• Raise your hand if you have any change in your pocket or purse. Record the number of raised hands.
CHAPTER 3 | PROBABILITY TOPICS 163
• Raise your hand if you rode a bus within the past month. Record the number of raised hands.
• Raise your hand if you answered "yes" to BOTH of the first two questions. Record the number of raised hands.
Use the class data as estimates of the following probabilities. P(change) means the probability that a randomly chosen
person in your class has change in his/her pocket or purse. P(bus) means the probability that a randomly chosen person
in your class rode a bus within the last month and so on. Discuss your answers.
• Find P(change).
• Find P(bus).
• Find P(change AND bus). Find the probability that a randomly chosen student in your class has change in his/her
pocket or purse and rode a bus within the last month.
• Find P(change|bus). Find the probability that a randomly chosen student has change given that he or she rode a
bus within the last month. Count all the students that rode a bus. From the group of students who rode a bus,
count those who have change. The probability is equal to those who have change and rode a bus divided by those
who rode a bus.
3.1 | Terminology
Probability is a measure that is associated with how certain we are of outcomes of a particular experiment or activity.
An e ...
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.
3 PROBABILITY TOPICSFigure 3.1 Meteor showers are rare, .docxtamicawaysmith
3 | PROBABILITY TOPICS
Figure 3.1 Meteor showers are rare, but the probability of them occurring can be calculated. (credit: Navicore/flickr)
Introduction
Chapter Objectives
By the end of this chapter, the student should be able to:
• Understand and use the terminology of probability.
• Determine whether two events are mutually exclusive and whether two events are independent.
• Calculate probabilities using the Addition Rules and Multiplication Rules.
• Construct and interpret Contingency Tables.
• Construct and interpret Venn Diagrams.
• Construct and interpret Tree Diagrams.
It is often necessary to "guess" about the outcome of an event in order to make a decision. Politicians study polls to guess
their likelihood of winning an election. Teachers choose a particular course of study based on what they think students can
comprehend. Doctors choose the treatments needed for various diseases based on their assessment of likely results. You
may have visited a casino where people play games chosen because of the belief that the likelihood of winning is good. You
may have chosen your course of study based on the probable availability of jobs.
You have, more than likely, used probability. In fact, you probably have an intuitive sense of probability. Probability deals
with the chance of an event occurring. Whenever you weigh the odds of whether or not to do your homework or to study
for an exam, you are using probability. In this chapter, you will learn how to solve probability problems using a systematic
approach.
Your instructor will survey your class. Count the number of students in the class today.
• Raise your hand if you have any change in your pocket or purse. Record the number of raised hands.
CHAPTER 3 | PROBABILITY TOPICS 163
• Raise your hand if you rode a bus within the past month. Record the number of raised hands.
• Raise your hand if you answered "yes" to BOTH of the first two questions. Record the number of raised hands.
Use the class data as estimates of the following probabilities. P(change) means the probability that a randomly chosen
person in your class has change in his/her pocket or purse. P(bus) means the probability that a randomly chosen person
in your class rode a bus within the last month and so on. Discuss your answers.
• Find P(change).
• Find P(bus).
• Find P(change AND bus). Find the probability that a randomly chosen student in your class has change in his/her
pocket or purse and rode a bus within the last month.
• Find P(change|bus). Find the probability that a randomly chosen student has change given that he or she rode a
bus within the last month. Count all the students that rode a bus. From the group of students who rode a bus,
count those who have change. The probability is equal to those who have change and rode a bus divided by those
who rode a bus.
3.1 | Terminology
Probability is a measure that is associated with how certain we are of outcomes of a particular experiment or activity.
An e ...
This lesson aims to introduce to you the world of reasoning and logic. Reasoning is one vital skill in studying Mathematics and its vast landscape for learning. Even in real world setting, one can use reasoning to prove or
disprove concepts or information. As learner of this lesson, you are expected
to achieve the minimum competency for this topic which is to use inductive
or deductive reasoning in an argument.
The first of two workshops I've created for children in my class about probability. This is used as a rotation activity after I have done some teaching on it first.
Ideas for teaching chance, data and interpretation of dataJoanne Villis
These activities have been designed specifically for Year 3 students according to the Australian Curriculum guidelines. However, they can be adapted to meet other standards or year levels.
The University of Maine at Augusta Name ______.docxchristalgrieg
The University of Maine at Augusta Name: ________________________
Mathematics Department Date: _________________________
MAT 280 F16 Location:______________________
MAT 280
Exam 1 Chapter 1.1-1.6
Please answer the following questions. Part credit is possible if the work indicates an understanding of
the objective under investigation. Students may use their laptops, tablets, textbooks, notes, calculators
and scrap paper. If used, scrap paper should be turned in with the exam. Students may not use smart
phones. If available homework should be turned in with the exam.
Time: 2:45 If staff is available, extra time is permitted.
1. Find a proposition with the given truth table.
p q ?
T T F
T F T
F T T
F F T
2. Write the truth table for the proposition (r q) (p → r). Use as many columns as necessary.
Label each column.
p Ans: p q r
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
Name: ________________________
3. Find a proposition using only pq and the connective V with the given truth table.
p
q
?
T T F
T F F
F T F
F F T
4. Determine whether p (q r) is equivalent to q (p r). Use as many columns as necessary.
Label each used column. Credit will only be when a completed truth table accompanies the answer.
p Ans: p q r
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
Name: ________________________
5. Write a proposition equivalent to ( p q) using only pq and the connective . . Support your
answer with a truth table. If necessary, insert columns in the truth table below to support your
answer
p
q
T T
T F
F T
F F
6. Prove that q p and its contrapositive are logically equivalent. If they are not equivalent, explain
why. Do the same for its inverse.
p
q
p
q
T T T T
T F T F
F T F T
F F F F
7. In the questions below write the statement in the form “If …, then ….”
a. Whenever the temperature drops below 35 degrees, children should wear boots
during recess.
b. You have completed your program’s requirements only if you are eligible to graduate.
Name: ________________________
8. Write the contrapositive, converse, and inverse of the following:
If I have a valid passport, I will be able to travel to Cuba.
.
a. Contrapositive:
b. Converse:
c. Inverse:
9. How many rows are required to show the truth table for the following compound proposition?
(q r ) → (p s) ...
1. Think “Relevant ==> Simple ==> Intricate.”
2. Visualize “mastery blocks.”
3. Generate comprehensive examples.
4. Assessment.
5. End with lead to next topic.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
This lesson aims to introduce to you the world of reasoning and logic. Reasoning is one vital skill in studying Mathematics and its vast landscape for learning. Even in real world setting, one can use reasoning to prove or
disprove concepts or information. As learner of this lesson, you are expected
to achieve the minimum competency for this topic which is to use inductive
or deductive reasoning in an argument.
The first of two workshops I've created for children in my class about probability. This is used as a rotation activity after I have done some teaching on it first.
Ideas for teaching chance, data and interpretation of dataJoanne Villis
These activities have been designed specifically for Year 3 students according to the Australian Curriculum guidelines. However, they can be adapted to meet other standards or year levels.
The University of Maine at Augusta Name ______.docxchristalgrieg
The University of Maine at Augusta Name: ________________________
Mathematics Department Date: _________________________
MAT 280 F16 Location:______________________
MAT 280
Exam 1 Chapter 1.1-1.6
Please answer the following questions. Part credit is possible if the work indicates an understanding of
the objective under investigation. Students may use their laptops, tablets, textbooks, notes, calculators
and scrap paper. If used, scrap paper should be turned in with the exam. Students may not use smart
phones. If available homework should be turned in with the exam.
Time: 2:45 If staff is available, extra time is permitted.
1. Find a proposition with the given truth table.
p q ?
T T F
T F T
F T T
F F T
2. Write the truth table for the proposition (r q) (p → r). Use as many columns as necessary.
Label each column.
p Ans: p q r
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
Name: ________________________
3. Find a proposition using only pq and the connective V with the given truth table.
p
q
?
T T F
T F F
F T F
F F T
4. Determine whether p (q r) is equivalent to q (p r). Use as many columns as necessary.
Label each used column. Credit will only be when a completed truth table accompanies the answer.
p Ans: p q r
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
Name: ________________________
5. Write a proposition equivalent to ( p q) using only pq and the connective . . Support your
answer with a truth table. If necessary, insert columns in the truth table below to support your
answer
p
q
T T
T F
F T
F F
6. Prove that q p and its contrapositive are logically equivalent. If they are not equivalent, explain
why. Do the same for its inverse.
p
q
p
q
T T T T
T F T F
F T F T
F F F F
7. In the questions below write the statement in the form “If …, then ….”
a. Whenever the temperature drops below 35 degrees, children should wear boots
during recess.
b. You have completed your program’s requirements only if you are eligible to graduate.
Name: ________________________
8. Write the contrapositive, converse, and inverse of the following:
If I have a valid passport, I will be able to travel to Cuba.
.
a. Contrapositive:
b. Converse:
c. Inverse:
9. How many rows are required to show the truth table for the following compound proposition?
(q r ) → (p s) ...
1. Think “Relevant ==> Simple ==> Intricate.”
2. Visualize “mastery blocks.”
3. Generate comprehensive examples.
4. Assessment.
5. End with lead to next topic.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
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.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Overview on Edible Vaccine: Pros & Cons with Mechanism
Lesson 2.3 Mathematical Reasoning and Logic
1.
2. Learning Outcomes:
At the end of this lesson, the students will be able to:
1. Analyze problems using different types of reasoning.
2. Apply different types of reasoning to justify
statements and arguments made about mathematics
and mathematical concepts.
3.
4. INDUCTIVE REASONING
It is the process of reaching a
general conclusion by
examining specific examples.
6. Example 1:
Use inductive reasoning to predict the next number in
each item.
1. 2,8,14,20,26, _____
2. 1,2,5,10,17,26, _____
7. Example 2:
A. Every sports car I have ever seen is red. Thus, all
sports cars are red.
B. The coin I pulled from the bag is a 5-peso coin.
Another 5-peso coin is drawn from the bag. A third
coin from the bag is again a 5-peso coin. Therefore,
all the coins in the bag are 5-peso coins.
8. Example 3:
Consider the following. Pick a number. Multiply the
number by 4, add 8 to the product, divide the sum by 2,
and subtract 5. Complete the above procedure for
several different numbers. Use inductive reasoning to
make a conjecture about the relationship between the
size of the resulting number and the size of the original
number.
9. Solution:
Suppose we start with seven as the original number. Then
repeat the process for different numbers. The procedure yields
the following:
We conjecture that the given procedure produces a number
that is one less than twice the original number.
10. Remarks:
When we use inductive reasoning, we have no
guarantee that our conclusion is correct. Just because
a pattern is true for a few cases, it does not mean the
pattern will continue. A statement is a true statement
provided that it is valid in all cases. If we can find one
case for which a statement is not valid, called a
counterexample, then it is a false statement.
11. DEDUCTIVE REASONING
It is the process of reaching a
conclusion by applying
general assumptions, procedures,
or principles.
12. DEDUCTIVE REASONING
Deduction starts out with a general
statement, or hypothesis, and
examines the possibilities to reach a
specific, logical conclusion.
13. Example 1:
1. All men are mortal. Kahwi is a man. Therefore,
Kahwi is mortal.
2. Corresponding parts of congruent triangles are
congruent. Triangle ABC is congruent to triangle
DEF. Angle B and angle E are corresponding angles.
Thus, angle B is congruent to angle E.
15. Determine if each of the following statement
uses inductive or deductive reasoning.
1. Teacher Erica is an enthusiastic and passionate teacher.
Therefore, all teachers are enthusiastic and passionate.
2. All dogs are animals. Dhai is a dog. Thus, Dhai is an
animal.
3. I got low score on the first long exam. I just recently took
the second long exam and I got low score. Therefore, I will
also get a low score on the third long exam.
IR
DR
IR
16. Determine if each of the following statement
uses inductive or deductive reasoning.
4. My classmates are disrespectful toward our instructor.
Hence, all students are disrespectful.
5. Last Wednesday it was raining. Today is Wednesday and it
is raining. Therefore, on the next Wednesday, it will also rain.
6. For any right triangle, the Pythagorean Theorem holds.
ABC is a right triangle, therefore for ABC the Pythagorean
Theorem holds.
7. All basketball players in your school are tall, so all
basketball players must be tall.
IR
IR
DR
IR
17. A logic puzzle is a puzzle deriving from the
mathematics field of deduction.
Logic puzzles can be solved by using deductive
reasoning and by organizing the data in a
given situation.
18. A logic puzzle is basically a description of an
event or any situation. Using the clues
provided, one has to piece together what
actually happened. This involves clear and
logical thinking, hence the term “logic” puzzles.
19. Example 1
Three musicians appeared at a concert. Their last names were Benton, Lanier, and
Rosario. Each plays only one of the following instruments: guitar, piano, or saxophone.
1. Benton and the guitar player arrived at the concert together.
2. The saxophone player performed before Benton.
3. Rosario wished the guitar player good luck.
Who played each instrument?
Guitar Piano Saxophone
Benton
Lanier
Rosario
20. Example 2
You have a basket containing ten apples. You have 10 friends,
who each desire an apple. You give each of your friends, one
apple. Now all your friends have one apple each, yet there is an
apple remaining in the basket. How?
Solution:
✎You give an apple to your first nine friends, and a basket with an apple
to your tenth friend. Each friend has an apple, and one of them has it
in a basket.
✎ Alternative answer: one friend already had an apple and put it in the
basket.
21. Example 3
A census-taker knocks on a door, and asks the woman inside
how many children she has and how old they are. “I have three
daughters, their ages are whole numbers, and the product of
their ages is 36,” says the mother. “That’s not enough
information”, responds the census-taker. “I’d tell you the sum of
their ages, but you’d still be stumped.” “I wish you’d tell me
something more.” “Okay, my oldest daughter Annie likes dogs.”
What are the ages of the three daughters (Zeitz, 2007)?
22. Solution:
After the first reading, it seems impossible- there isn’t enough information to
determine the ages. The product of the ages is 36, so there are only a few possible
triples of ages. Here is a table of all the possibilities
Age 1, 1,36 1, 2,18 1, 3, 12 1, 4, 9 1, 6, 6 2, 2, 9 2, 3, 6 3, 3, 4
Sum 38 21 16 14 13 13 11 10
Now we see what is going on. The mother’s second statement (“I’d tell you the
sum of their ages, but you’d still be stumped.) gives valuable information. It tells
that the ages are either 1,6,6 or 2,2,9, for in all other cases, knowledge of the sum
would tell unambiguously what the ages are. The final clue now makes sense, it
tells that there is an oldest daughter, eliminating 1, 6,6. The daughters are thus 2,
2, and 9 years old.
Notas del editor
Reasoning is a process based on experience and principles that allow one to arrive at a conclusion.
Reasoning is a process based on experience and principles that allow one to arrive at a conclusion.
We make generalizations from the part to the whole. If we are not careful, it can lead to erroneous or mistake conclusions:
Conjecture- estimation or guess
Each successive number is 6 larger than the preceding number.
It is not enough that the deduction is logically sound; the assumption (1) must also be true. Consider the following: "1) All cats are red. 2) Kitty is a cat, therefore Kitty is red." It is logically valid but leads to a non-valid conclusion because not 'all cats are red'.
In mathematics, deductive reasoning makes use of definitions, axioms, theorems and rules and inference.