2.4 Deductive Reasoning Inductive Reasoning: using patterns and examples to draw conclusions Deductive Reasoning: (logical reasoning) is the process of reasoning logically from given statements to a conclusion.
Inductive or Deductive?
Every time Kyle skips breakfast he has a head ache later that morning.
Kyle makes sure to eat breakfast to prevent getting a head ache later.
Marcus knows that if he seen speeding by police then he will get a speeding ticket.
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Law of Detachment
Example 1: An auto mechanic knows that if a car has a dead battery, the car will not start. A mechanic begins work on a car and finds the battery is dead. What conclusion can she make?
In example 1 the mechanic is using the law of detachment Law of Detachment: If p q is true and p is true, then q is true.
A gardener knows that if it rains, the garden will be watered. It is raining. What conclusion can he make? 2
Using the Law of Detachment
(More difficult of the two laws)
Example 1: Example 2: If a baseball player is a pitcher, then that player should not pitch a complete game two days in a row. Vladimir Nunez is a pitcher. On Monday, he pitches a complete game. What can you conclude. 3
Law of Syllogism Law of Syllogism: allows you to state a conclusion from 2 true statements when the conclusion of one statement is the hypothesis of the other statement.
If p q and q r are true statements, then p r is a true statement.
Example 1: Use the Law of Syllogism to draw a conclusion from the following true statements If a number is prime, then it does not have repeated factors.
If a number does not have repeated factors, then it is not a perfect square.
Example 2: Use the Law of Syllogism to draw a conclusion from the following true statements
If a quadrilateral is a square, then it contains four right angles.
If a quadrilateral contains four right angles, then it is a rectangle.
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Example 3: If possible, state a conclusion using the Law of Syllogism. If it is not possible to use this law explain why.
a. If a number ends in 0, then it is divisible by 10.
If a number is divisible by 10, then it is divisible by 5.
b. If a number ends in 6, then it is divisible by 2.
If a number ends in 4, then it is divisible by 2.
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Using Both Laws Together Example 1: Use both laws to determine a conclusion
If a river is more than 4000 mi long, then it is longer than the Amazon.
If a river is longer than the Amazon, then it is the longest river in the world.
The Nile is 4132 mi long.
Example 2: Use both laws to determine a conclusion
The Volga River is in Europe.
If a river is less than 2300 mi long, it is not one of the world's ten longest rivers.
If a river is in Europe, then it is less than 2300 mi long.
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2.5 Postulates and Paragraph Proofs What is a postulate?
What is an axiom?
Theorem: a statement we can prove using definitions, postulates and properties.
Proof: logical argument use to prove a statement is true
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p. 127 in your book
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Take a slip of paper (I will give you one)
Explain why:
If M is the midpoint of XY, then XM = MY.
Write your explanation in a short paragraph (2 or 3 sentences).
You will turn this in.
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Steps to a Proof:
1) List the given information.
2) List the statement you are proving
3) Create an argument (a logical chain of statements from the given info to the statement you are proving)
4) Prove reasons for your statements
(definitions, algebraic properties, postulates)
5) State what you have proven
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This is also in your book on page 129.
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Homework:
LOD and LOS Practice
p. 131 # 27­29
Write a paragraph proof for:
Given:
2 and 4 are vertical angles Prove: m 2 = m
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