Transcript 2-3: Deductive Reasoning

2-3: Deductive
Reasoning
Expectations:
L3.1.1: Distinguish between inductive and deductive
reasoning, identifying and providing examples of each.
L3.1.3: Define and explain the roles of axioms (postulates),
definitions, theorems, counterexamples, and proofs in the
logical structure of mathematics. Identify and give examples
of each.
L3.3.3: Explain the difference between a necessary and a
sufficient condition within the statement of a theorem.
Determine the correct conclusions based on interpreting a
theorem in which necessary or sufficient conditions in the
theorem or hypothesis are satisfied.
4/7/2016
2-3: Deductive Reasoning
1
Diagonals
A segment is a diagonal of a polygon iff its
endpoints are 2 non-consecutive vertices
of a polygon.
ex: AC, BE and DF are diagonals for
polygon ABCDEF.
A
F
B
E
C
4/7/2016
D
2-3: Deductive Reasoning
2
Use a pattern to answer the
question.
How many diagonals does an octagon
have?
4/7/2016
2-3: Deductive Reasoning
3
Patterns are not proof – they are conjecture.
Remember, this is inductive reasoning
which is not valid for making a proof.
The following slides give us some properties
of deductive reasoning which is valid for
proving statements true.
4/7/2016
2-3: Deductive Reasoning
4
Law of Detachment
If p => q is true and p is a true statement,
then ___ must be true.
ex:
1. If today is Monday, then tomorrow is
Tuesday.
2. Today is Monday.
Conclude: _________________.
4/7/2016
2-3: Deductive Reasoning
5
Make a conclusion based on the following
true statements.
a. If the air conditioner is on, then it is hot
outside.
b. The air conditioner is on.
4/7/2016
2-3: Deductive Reasoning
6
Make a conclusion based on the following
true statements.
a. If it is raining, then it is humid.
b. It is humid.
4/7/2016
2-3: Deductive Reasoning
7
If p => q and q are true
_________________ can be made.
This is referred to as affirming the
consequent.
4/7/2016
2-3: Deductive Reasoning
8
Necessary and Sufficient
Conditions
In the statement of a theorem in “if- then”
form, we can talk about sufficient
conditions for the truth of the statement
and necessary conditions of the truth of
the statement.
This is really just another way of looking at
the Law of Detachment and Affirming the
Consequent.
4/7/2016
2-3: Deductive Reasoning
9
The ___________ is a sufficient condition
for the conclusion and the conclusion is a
_____________ condition of the
hypothesis.
4/7/2016
2-3: Deductive Reasoning
10
Necessary
Consider the statement p => q. We say q is
a necessary condition for (or of) p.
Ex: “If if is Sunday, then we do not have
school.”
A necessary condition of it being Sunday is
that we do not have school.
4/7/2016
2-3: Deductive Reasoning
11
Sufficient Condition
A sufficient condition is a condition that all
by itself guarantees another statement
must be true.
Ex: If you legally drive a car, then you are at
least 15 years old.”
Driving legally guarantees that a person
must be at least 15 years old.
4/7/2016
2-3: Deductive Reasoning
12
Notice that, “We do not have school today”
is not sufficient to guarantee that today is
Sunday.
4/7/2016
2-3: Deductive Reasoning
13
“If M is the midpoint of segment
AB, then AM ≅ MB.”
Given that M is the midpoint, it is necessary
(true) that AM ≅ MB.
This means that M being the midpoint is a
____________ condition for AM  MB.
4/7/2016
2-3: Deductive Reasoning
14
Notice simply saying AM ≅ MB does not
guarantee that M is the midpoint of AB, so
it is not a sufficient condition.
4/7/2016
2-3: Deductive Reasoning
15
“If a triangle is equilateral, then it is
isosceles.”
A triangle having 3 congruent sides
(equilateral) guarantees that at least 2
sides are congruent, so a triangle being
equilateral is sufficient to say it is
isosceles.
4/7/2016
2-3: Deductive Reasoning
16
“If a person teaches mathematics,
then they are good at algebra.”
Because Trevor is a math teacher, can we
conclude he is good at algebra. Justify your
answer.
4/7/2016
2-3: Deductive Reasoning
17
“If a person teaches mathematics,
then they are good at algebra.”
Betty is 32 and is very good at algebra. Can we
correctly conclude that she is a math teacher?
Justify.
4/7/2016
2-3: Deductive Reasoning
18
Bi-Conditional Statements
If a statement and its converse are both true
it is called a bi-conditional statement and
can be written in ________________
form.
4/7/2016
2-3: Deductive Reasoning
19
Ex:
“If an angle is a right angle, then its measure
is exactly 90°” and “If the measure of an
angle is exactly 90°, then it is a right
angle” are true converses of each other so
they can be combined into a single
statement.
__________________________________
__________________________________
4/7/2016
2-3: Deductive Reasoning
20
Necessary and Sufficient
If a statement is a bi-conditional statement
then either part is a necessary and
sufficient condition for the entire
statement.
Remember all definitions are bi-conditional
statements.
4/7/2016
2-3: Deductive Reasoning
21
A triangle is a right triangle iff it has a right
angle.
Being a right triangle is necessary and
sufficient for a triangle to have a right
angle and possessing a right angle is
necessary and sufficient for a triangle to
be a right triangle.
4/7/2016
2-3: Deductive Reasoning
22
Necessary, Sufficient, Both or
Neither
Given the true statement:
“If a quadrilateral is a rhombus, then its
diagonals are perpendicular.”
Is the following statement necessary,
sufficient, both or neither?
The diagonals of ABCD are perpendicular.
4/7/2016
2-3: Deductive Reasoning
23
Which of the following is a sufficient but NOT
necessary condition for angles to be
supplementary?
A. they are both acute angles.
B. they are adjacent
C. their measures add to 90.
D. they are coplanar.
E. they form a linear pair.
4/7/2016
2-3: Deductive Reasoning
24
Necessary, Sufficient, Both or
Neither
Given the true statement:
“A quadrilateral is a rhombus if and only if its
4 sides are congruent.”
Is the following statement necessary,
sufficient, both or neither?
The sides of ABCD are all congruent.
4/7/2016
2-3: Deductive Reasoning
25
Law of Syllogism: Transitive Law of
Logic (A Form of Logical Argument)
If p => q and q =>r, then __________.
ex:
1. If a polygon is a square, then it is a
rhombus.
2. If a polygon is a rhombus, then it is a
parallelogram.
Conclude: __________________________
__________________________________.
4/7/2016
2-3: Deductive Reasoning
26
Make a conclusion based on the following.
a. If a quadrilateral is a square, then it has
4 right angles.
b. If a quadrilateral has 4 right angles, then
it is a rectangle.
4/7/2016
2-3: Deductive Reasoning
27
Assignment
pages 89 - 91,
# 18 - 30(evens), 42, 44 and 46
4/7/2016
2-3: Deductive Reasoning
28