Transcript What is Deductive Reasoning?

Use Inductive Reasoning
Objectives
1. To form conjectures through inductive
reasoning
2. To disprove a conjecture with a
counterexample
3. To avoid fallacies of inductive reasoning
Example 1
You’re at school eating lunch. You ingest some air
while eating, which causes you to belch.
Afterward, you notice a number of students
staring at you with disgust. You burp again, and
looks of distaste greet your natural bodily
function. You have similar experiences over the
course of the next couple of days. Finally, you
conclude that belching in public is socially
unacceptable. The process that lead you to this
conclusion is called inductive reasoning.
Inductive Reasoning
Inductive reasoning
is the process of
observing data,
recognizing
patterns, and
making
generalizations
based on your
observations.
Generalization
Generalization:
statement that
applies to every
member of a group
• Science =
hypothesis
• Math = conjecture
Conjecture
A conjecture is a
general, unproven
statement believed
to be true based on
investigation or
observation
Inductive Reasoning
Inductive reasoning can
be used to make
predictions about the
future based on the
past or to make
conjectures about the
past based on the
present.
Example 2
A scientist takes a piece of salt, turns it over
a Bunsen burner, and observes that it
burns with a yellow flame. She does this
with many other pieces of salt, finding they
all burn with a yellow flame. She therefore
makes the conjecture: “All salt burns with
a yellow flame.”
Inductive Reasoning
Inductive Reasoning
Example 6
(An allegory) Student A neglected to do his/her
homework on numerous occasions. When Student
A's mean teacher popped a quiz on the class,
Student A failed. After the quiz, Student A had
several other HW assignments that he/she also
neglected to complete. When test time rolled
around, Student A failed the exam . Students B-F
behaved in a similar, academically deplorable
manner. Use inductive reasoning to make a
conjecture about the relationship between homework
and test/quiz performance.
Example 7
Inductive reasoning does not
always lead to the truth.
What are some famous
examples of conjectures
that were later discovered to
be false?
To Prove or To Disprove
In science, experiments are used to prove or
disprove an hypothesis.
In math, deductive reasoning is used to
prove conjectures and counterexamples
are used to disprove them.
Counterexample
A counterexample is a single case in which
a conjecture is not true.
Example 8
On her first road trip, Little Window Watcher Wilma
observes a number of vehicles. Each one she
observes has four wheels. She conjectures “All
vehicles have four wheels.” What is wrong with
her conjecture? What counterexample will
disprove it?
Conjecture: All
vehicles have 4
wheels
Example 10
Kenny makes the following conjecture about
the sum of two numbers. Find a
counterexample to disprove Kenny’s
conjecture.
Conjecture: The sum of two numbers is
always greater than the larger number.
Example 11
Joe has a friend who just happens to be a
Native American named Victor. One day
Victor gave Joe a CD. The next day Victor
decided that he wanted the CD back, and
so he confronted Joe. After reluctantly
giving the CD back to his friend, Joe made
the conjecture: “Victor, like all Native
Americans, is an Indian Giver.” What is
wrong with his conjecture? What does this
example illustrate?
Inductive Fallacies
The previous example illustrated an inductive
fallacy, where a reliable conjecture cannot be
justifiably made. Joe was guilty of a Hasty
Generalization, basing a conclusion on too little
information. Here are some others:
• Unrepresentative Sample
• False Analogy
• Slothful Induction
• Fallacy of Exclusion
Inductive Fallacies
As a group, match each inductive fallacy
definition with the corresponding example.
Be sure to take some notes, as this
priceless information is not in your
textbook.
Apply Deductive Reasoning
Objectives:
1. To recognize deductive reasoning and
use it to arrive at a true conclusion.
History
When the architects designed
this school building, they
were approached by an
ancient secret society whose
members make up numerous
Texas dignitaries. They
convinced the architects to
add several secret passages
and hidden conference rooms
to their design plans.
History
Sometimes when I stay
after school late into
the evening grading
papers, planning
lessons, and contacting
parents, I hear strange
and inauspicious
sounds emanating from
behind one of my walls.
Conspiracy?
Thus, it is my conjecture that one of the secret
society's hidden passages lies between the
walls of Room D202 and D204. This is a bold
and perhaps conspiratorial conjecture, but I
am confident that it is true. (You should hear
the sounds--Oh, my!)
Stop Making Fun of Me!
I have told few people of
my theory, and they
unanimously dismiss
my conviction with
ridicule. (Then they
ask me if I frequently
watch re-runs of the XFiles with the notion
that the story lines are
largely nonfiction!)
Redemption
To convince the skeptics and to redeem my
reputation, I need absolute and conclusive
proof that there exists a hidden passage
between these classrooms.
Proof
The Principle of
Laplace:
The weight of
evidence for an
extraordinary claim
must be
proportional to its
strangeness.
In Other Words…
“Extraordinary claims
require
extraordinary
evidence.”
-Carl Sagan
Example 1
In your group, come up with a nondestructive
method for proving or disproving the
extraordinary claim that there’s a secret
tunnel between D202 and D204.
Example 2
In the Sudoku
puzzle shown,
what number
must be written
in the blue box?
Why?
?
Deductive Reasoning
The process of
demonstrating that
if certain
statements are
accepted as true,
then other
statements can be
shown to follow
from them.
Deductive Reasoning
The “accepted” statements are sometimes
premises or assumptions, and all
deductive arguments must have them.
Deductive reasoning uses logical inference
to build on these assumptions.
Unlike inductive reasoning, deductive
reasoning will always lead to the truth
as long as the assumptions are true.
Example 2
All humans have
skeletons is a
reasonable
assumption. So,
since Mr. Asake is
a human, what
must be true about
him?
Deductive Reasoning
Deductive Reasoning
Inductive vs. Deductive
1. We use inductive reasoning to investigate
and discover things about our world.
2. Since the conjectures we make using our
inductive reasoning is based on our fallible
observation skills, we can be wrong.
3. We can search for a counterexample to
disprove our conjectures.
4. In mathematics, we use our deductive
reasoning to prove our conjectures beyond
all uncertainty.
Flavors of Deductive Reasoning
Deductive reasoning comes in a variety of
flavors, and just to make things confusing,
each flavor is know by a number of
different names.
1. Law of Detachment = Modus Ponens =
Affirming the Antecedent
2. Denying the Consequent = Modus Tollens
3. Law of Syllogism = Chain Rule
Law of Detachment
Symbols
Example
pq
If Watson had chalk on his fingers,
then he had been playing billiards.
p
Watson had chalk between his fingers
upon returning from the club.
q
Therefore Watson had been playing
billiards.
Denying the Consequent
Symbols
Example
pq
If Watson wished to invest his money in S. African
securities with Thurston, then he would have had his
check book when playing billiards with Thurston.
~q
Watson did not have his checkbook when he played
billiards with Thurston.
 ~ p
Therefore Watson did not wish to invest his money
in S. African securities with Thurston.
Law of Syllogism
Symbols
Example
pq
If I eat pizza after midnight, then I will
have nightmares.
qr
If I have nightmares, then I will get very
little sleep.
pr
Therefore, if I eat pizza after midnight,
then I will get very little sleep.
Example 5
Use one of the laws of deductive reasoning
to make a valid conclusion.
If two segments have the same length, then
they are congruent. You know that
BC = XY.
Example 6
Use one of the laws of deductive reasoning
to make a valid conclusion.
If x2 > 25, then x2 > 20.
If x > 5, then x2 > 25.
Example 7
Use one of the laws of deductive reasoning
to make a valid conclusion.
If a polygon is regular, then it is both
equilateral and equiangular.
Pentagon ABCDE is not equilateral or
equiangular.
Use Postulates & Diagrams
Objectives:
1. To illustrate and understand postulates
about lines and planes
2. To accurately interpret geometric
diagrams
3. To use properties of special pairs of
angles to find angle measurements
Example 1
What is the length of
S
SM
?
A
6 cm
20 cm
M
Example 1
You basically used the Segment Addition
Postulate to get the length of the segment,
where SA + AM = SM.
S
A
26 cm
M
Postulates
As you build a
deductive system
like geometry, you
demonstrate that
certain statements
are logical
consequences of
other previously
accepted or proven
statements.
Postulates
This chain of logical
reasoning must
begin somewhere,
so every deductive
system must contain
some statements
that are never
proved. In
geometry, these are
called postulates.
Postulates and Theorems
• Postulates are
statements in
geometry that are so
basic, they are
assumed to be true
without proof.
– Sometimes called
axioms.
• Theorems are
statements that were
once conjectures but
have since been
proven to be true
based on postulates,
definitions, properties,
or previously proven
conjectures.
Both postulates and theorems are ordinarily written in conditional form.
Repeat Times Seven!
Example 1
State the postulate illustrated by the
diagram.
Example 2
How does the
diagram shown
illustrate one or
more
postulates?
Interpreting Diagrams
When you interpret a
diagram, you can
assume only
information about
size or measure if it
is marked.
Interpreting Diagrams
Interpreting Diagrams
Example 3
Sketch and carefully label a diagram with
plane A containing noncollinear points R,
O, and W, and plane B containing
noncollinear points N, W, and R.
Perpendicular Figures
A line is perpendicular
to a plane if and only
if the line intersects
the plane in a point
and is perpendicular
to every line in the
plane that intersects
it at that point.
Example 4
Which of the following
cannot be assumed
from the diagram?
1. A, B, and F are
collinear.
2. E, B, and D are
collinear.
3. AB  plane S
Example 4
Which of the following
cannot be assumed
from the diagram?
4. CD  plane T
5. AF intersects BC at
point B.
Example 5a
1. Identify all
linear pairs of
angles.
2. Identify all
pairs of vertical
angles.
2
3
1
4
Example 5b
3. If m<1 = 40°,
find the
measures of
the other
angles in the
diagram.
2
3
1
4
Click me!
Linear Pair Postulate
If two angles form a linear pair, then they are
supplementary.
Do we have to prove this?
Vertical Angle Congruence Theorem
Vertical angles are congruent.
Example 6
Find the missing measure of each angle.
60
65
Example 7
Find the value of x and y.
3y - 1
2x +5
35
Example 8
Find the value(s) of x.
Example 9: SAT
For the two intersecting lines, which of the
following must be true?
I. a > c
II. a = 2b
III. a + 60 = b + c
a
60
b
c
Example 10: SAT
In the figure, what is
the value of y?
x
3x
y
2x