Transcript Logic - Denise Kapler

Logic
Logic
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Logical progression of thought
A path others can follow and agree with
Begins with a foundation of accepted
In Euclidean Geometry begin with point, line
and plane
Short sweet
and to the point
Number Pattern
Is this proof of how
numbers were developed?
Mathematical
Proof
2=1
a=b
a2 = ab
a2 - b2 = ab-b2
(a-b)(a+b) = b(a-b)
a+b = b
b+b = b
2b = b
2=1
Geometry
Undefined terms
• Are not defined, but instead explained.
• Form the foundation for all definitions in geometry.
Postulates
• A statement that is accepted as true without proof.
Theorem
• A statement in geometry that has been proved.
Inductive Reasoning
•A form of reasoning that draws a conclusion
based on the observation of patterns.
•Steps
1. Identify a pattern
2. Make a conjecture
•Find counterexample to disprove conjecture
Inductive Reasoning
•Does not definitely prove a statement,
rather assumes it
•Educated Guess at what might be true
Example
Polling
30% of those polled agree therefore 30% of
general population
Inductive Reasoning
Not Proof
Identifying a Pattern
Find the next item in the pattern.
7, 14, 21, 28, …
Multiples of 7 make up the pattern.
The next multiple is 35.
Identifying a Pattern
Find the next item in the pattern.
4, 9, 16, …
Sums of odd numbers make up the pattern.
1=1
The next number is 25.
1+ 3=4
1+ 3 +5=9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
Identifying a Pattern
Find the next item in the pattern.
In this pattern, the figure rotates 90° counter-clockwise
each time.
The next figure is
.
Making a Conjecture
Complete the conjecture.
The sum of two odd numbers is ? .
List some examples and look for a pattern.
1+1=2
3.14 + 0.01 = 3.15
3,900 + 1,000,017 = 1,003,917
The sum of two positive numbers is positive.
Identifying a Pattern
Find the next item in the pattern.
January, March, May, ...
Alternating months of the year make up the pattern.
The next month is July.
The next month is…
then August
Perhaps the pattern was…
Months with 31 days.
Complete the conjecture.
The product of two odd numbers is ? .
List some examples and look for a pattern.
11=1
33=9
5  7 = 35
The product of two odd numbers is odd.
Inductive Reasoning
Counterexample - An example which disproves
a conclusion
•Observation
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 are odd
•Conclusion
All prime numbers are odd.
2 is a counterexample
Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
For every integer n, n3 is positive.
Pick integers and substitute them into the expression to see if the conjecture holds.
Let n = 1. Since n3 = 1 and 1 > 0, the conjecture holds.
Let n = –3. Since n3 = –27 and –27  0, the conjecture is false.
n = –3 is a counterexample.
Inductive Reasoning
Example 1
90% of humans are right-handed.
Joe is a human. Therefore, the probability that Joe is right-handed is 90%.
Example 2
Every life form that everyone knows of
depends on liquid water to exist.
Therefore, all known life depends on liquid water to exist.
Example 3
All of the swans that all living beings have ever
seen are white. Therefore, all swans are white.
Inductive reasoning allows for the possibility that
the conclusion is false,
even where all of the premises are true
Conjectures about our class….
True?
Supplementary angles are adjacent.
23°
157°
The supplementary angles are not adjacent, so the conjecture is false.
Homework 2.1 and 2.2
To determine truth in geometry…
Information is put into a conditional statement.
The truth can then be tested.
A conditional statement in math is a statement in
the if-then form.
If hypothesis, then conclusion
A bi-conditional statement is of the form If and
only if.
If and only if hypothesis, then conclusion.
Underline the hypothesis twice
The conclusion once
1. A figure is a parallelogram if it is a rectangle.
2. Four angles are formed if two lines intersect.
Analyzing the Truth Value of a
Conditional Statement
Determine if the conditional is true. If false,
give a counterexample.
If two angles are acute, then they are congruent.
You can have acute angles with measures of
80° and 30°. In this case, the hypothesis is
true, but the conclusion is false.
Since you can find a counterexample, the
conditional is false.
Analyzing the Truth Value of a
Conditional Statement
Determine if the conditional is true.
“If a number is odd, then it is divisible by 3”
If false, give a counterexample.
An example of an odd number is 7. It is not
divisible by 3. In this case, the hypothesis is true,
but the conclusion is false. Since you can find a
counterexample, the conditional is false.
For Problems 1 and 2: Identify the hypothesis
and conclusion of each conditional.
1. A triangle with one right angle is a right triangle.
H: A triangle has one right angle.
C: The triangle is a right triangle.
2. All even numbers are divisible by 2.
H: A number is even.
C: The number is divisible by 2.
3. Determine if the statement “If n2 = 144, then
n = 12” is true. If false, give a counterexample.
False; n = –12.
Identify the hypothesis and conclusion of each conditional.
1. A mapping that is
2. The
a reflection is a type of transformation.
H: A mapping is a reflection.
C: The mapping is a transformation.
quotient of two negative numbers is positive.
H: Two numbers are negative.
C: The quotient is positive.
3. Determine if
the conditional “If x is a number then |x| > 0” is true.
If false, give a counterexample.
False; x = 0.
Different Forms of Conditional Statements
Given Conditional Statement
If an animal is a cat, then it has four paws.
Converse: If an animal has 4 paws, then it is a cat.
There are other animals that have 4 paws that are not
cats, so the converse is false.
Inverse: If an animal is not a cat, then it does not
have 4 paws.
There are animals that are not cats that have 4 paws,
so the inverse is false.
Contrapositive: If an animal does not have 4 paws,
then it is not a cat; True.
Cats have 4 paws, so the contrapositive is true.
A bi-conditional statement is of the form If and
only if.
If and only if hypothesis, then conclusion.
Example
A triangle is isosceles if and only if
the triangle has two congruent sides.
Write as a biconditional
Parallel lines are two coplanar lines
that never intersect
Two lines are parallel if and only if
they are coplanar and never intersect.
Homework 2.3
To determine truth in geometry…
Deductive Reasoning.
Beyond a shadow of a doubt.
Deductive reasoning
Uses logic to draw conclusions from
•Given facts
•Definitions
•Properties.
True or False
And how do you know?
A pair of angles is a linear pair.
The angles are supplementary angles.
Two angles are complementary and congruent.
The measure of each angle is 45 .
Modus Ponens
Most common deductive logical argument
p⇒q
p∴q
If p, then q
p, therefore q
Example
If I stub my toe, then I will be in pain.
I stub my toe.
Therefore, I am in pain.
Modus Tollens
Second form of deductive logic is
p⇒q
~q ∴ ~p
If p, then q
not q, therefore not p
Example
If today is Thursday, then the cafeteria will be
serving burritos.
The cafeteria is not serving burritos,
therefore today is not Thursday.
If-Then Transitive Property
Third form of deductive logic
A chains of logic where one thing implies another thing.
p⇒q
q⇒r∴p⇒r
If p, then q
If q, then r, therefore if p, then r
Example
If today is Thursday, then the cafeteria will be
serving burritos.
If the cafeteria will be serving burritos, then I will be
happy.
Therefore, if today is Thursday, then I will be happy.
Deductive reasoning
Three forms
p⇒q
p∴q
p⇒q
~q ∴ ~p
p⇒q
q⇒r∴p⇒r
Draw a conclusion from the given information.
If a polygon is a triangle, then it has three sides.
If a polygon has three sides, then it is not a quadrilateral.
Polygon P is a triangle.
Conclusion: Polygon P is not a quadrilateral.
Homework 2.4
Proof
1. Algebraic
2. Geometric
Proof - argument that uses
•Logic
•Definitions
•Properties, and
•Previously proven statements
to show that a conclusion is true.
An important part of writing a
proof is giving justifications to
show that every step is valid.
Algebraic Proof
• Properties of Real Numbers
Equality
Distributive Property
a(b + c) = ab + ac.
• Substitution
Practice Solving an Equation with Algebra
Solve the equation 4m – 8 = –12.
Write a justification for each step.
4m – 8 = –12
+8
+8
Given equation
Addition Property of Equality
4m
Simplify.
= –4
Division Property of Equality
m = –1
Simplify.
Practice Solving an Equation with Algebra
Solve the equation
.
Write a justification for each step.
Given equation
Multiplication Property of Equality.
t = –14
Simplify.
Solving an Equation with Algebra
Solve for x. Write a justification for each step.
NO = NM + MO
Segment Addition Post.
4x – 4 = 2x + (3x – 9) Substitution Property of Equality
4x – 4 = 5x – 9
–4 = x – 9
5=x
Simplify.
Subtraction Property of Equality
Addition Property of Equality
Homework 2.5
Algebraic Proof
Geometric Proof
 Prove geometric theorems by using
deductive reasoning.
 Two-column proofs.
Remember!
Numbers are equal (=) and
figures are congruent ().
When writing a proof:
1. Justify each logical step with a reason.
2. Each step must be clear enough so that anyone
who reads your proof will understand them.
Hypothesis
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Definitions
Postulates
Properties
Theorems
Conclusion
Proof Steps:
1. Start with given (hypothesis)
2. Logically connect given to conclusion
Progressives statements with reasons
3. End with conclusion
Two Column Proof – organizes your work
Statement
Reason
Writing Reasons Using a Two Column Proof
Write a reason for each
step, given that A and B
are supplementary and
mA = 45°.
1. A and B are supplementary.
mA = 45°
Given
2. mA + mB = 180°
Def. of supp s
3. 45° + mB = 180°
Subst. Prop of =
Steps 1, 2
Subtr. Prop of =
4. mB = 135°
Writing Reasons Using a Two Column Proof
Write a reason for
each step, given that
B is the midpoint of AC
and AB  EF.
1. B is the midpoint of AC.
Given
2. AB  BC
Def. of mdpt.
3. AB  EF
Given
4. BC  EF
Trans. Prop. of 
Completing a Two-Column Proof
Given: XY
Prove: XY  XY
Statements
1.
XY
2. XY = XY
3. XY
.

XY
Reasons
1. Given
2. Reflex.
.
Prop. of =
3. Def. of  segs.
Example 4
Completing a Two-Column Proof
Given: 1 and 2 are supplementary, and 1  3
Prove: 3 and 2 are supplementary.
Example 4 Continued
Statements
Reasons
1. 1 and 2 are supplementary. 1. Given
1  3
2. m1 + m2 = 180°
of supp. s
2. Def.
.
= m3
3. m1
.
3. Def. of  s
4. m3 + m2 = 180°
4. Subst.
5. 3 and 2 are supplementary 5. Def. of supp. s
Use a Two Column Proof
Given: 1, 2 , 3, 4
Prove: m1 + m2 = m1 + m4
1. 1 and 2 are supp.
1. Linear Pair Thm.
1 and 4 are supp.
2. m1 + m2 = 180°,
m1 + m4 = 180°
2. Def. of supp. s
3. m1 + m2 = m1 + m4 3. Subst.
Homework 2.6
Geometric Proof
There are nine
compositions (A to I)
of eight colored
cubes.
Find two identical
compositions.
They can be rotated.
Solution:
Compositions
D and I
are identical.
Four flat cubes
Their patterns are drawn with
bold black lines.
Which can be drawn without
taking your pencil off the paper or
going along the same line twice?
Which of them can't be drawn in
this way?
Shapes A and D
can be drawn
without taking
your pencil off
the paper or
going along the
same line
twice.
Shapes B and C
can't be drawn
in this way.