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S:\Prentice Hall Resouces\Math\Power Point\Math Topics 2\Revised Power Points\Lesson 25 Deductive Reasoning

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S:\Prentice Hall Resouces\Math\Power Point\Math Topics 2\Revised Power Points\Lesson 25 Deductive Reasoning

1. 1. <ul><li>OBJECTIVES : </li></ul><ul><li>Students will be able to arrive at a conclusion from premises that are given and accepted. </li></ul><ul><li>Students will be able to put several related conditional statements into logical order. </li></ul><ul><li>Students will be able to critique deductive arguments for validity. </li></ul>LESSON 25: Deductive Reasoning
2. 2. Vocabulary: LESSON 25: Deductive Reasoning A conditional statement is a statement in the if-then form where the if-clause = the hypothesis and the then-clause = the conclusion. (see pg. 203) A conclusion asserts that the general statement, which applies to the group, also applies to the individual case. (see pg. 202) Deductive reasoning is the process of arriving at a conclusion from premises that are given and accepted. (see pg. 202) A major premise is a general statement about an entire group. (see pg. 202) A minor premise is a specific statement indicating that an individual case belongs to the group. (see pg. 202) What is a conditional statement? See page 202 What is a conclusion? See page 202 What is deductive reasoning? See page 202 What is a major premise? See page 202 What is a minor premise? See page 202
3. 3. Vocabulary: LESSON 25: Deductive Reasoning A syllogism [sil-uh-jiz-uhm] is a simple form of deductive reasoning which combines a major premise and a minor premise to reach a conclusion. (see pg. 202) Valid reasoning arrives at conclusions that, based on the given premises and conditions, are absolutely true . (see pg. 206) Invalid reasoning arrives at conclusions that, based on the given premises and conditions, are false or not necessarily true . (see pg. 206) Critiquing is the process of evaluating deductive arguments for their validity. (see pg. 206) What is a syllogism? See page 202 What is valid reasoning? See page 206 What is invalid reasoning? See page 206 What is critiquing? See page 206
4. 4. LESSON 25: Deductive Reasoning Syllogism involve combining a major premise and a minor premise to reach a conclusion: All cars have 4 wheels Major Premise: General statement about an entire group Minor Premise: Specific statement indicating an individual case belongs to a group Example: This is an example of a major premise because it is a statement about an entire group all cars . Is this a major or minor premise? My Nissan Sentra is a car This is an example of a Minor premise because it is a statement about my Nissan Sentra Is this a major or minor premise? Therefore, a Nissan Sentra has four wheels This conclusion can be made by combining the major and minor premise.
5. 5. LESSON 25: Conditional Statements A conditional statement is written in the form if p , then q . If a given condition is met ( if p ), then another condition is true or an event will happen ( then q ). The if-clause is the hypothesis ; the then-clause is the conclusion . Conditional statements can be linked with minor premises to make conclusions. Example If tomorrow is Friday, then today is Thursday What is your hypothesis? See page 203 What is your conclusion? See page 203 Tomorrow is Friday, - Minor Premise Therefore, today is Thursday - Conclusion
6. 6. LESSON 25: Conditional Statements A conditional statement is written in the form if p , then q . If a given condition is met ( if p ), then another condition is true or an event will happen ( then q ). The if-clause is the hypothesis ; the then-clause is the conclusion . Conditional statements can be linked with minor premises to make conclusions. Example 2) If a Summit student is in class then it is a weekday What is your hypothesis? See page 203 What is your conclusion? See page 203 A Summit student is in class, - Minor Premise Therefore, it is a weekday - Conclusion Can someone think of another conditional statement?
7. 7. LESSON 25: Logical Order of Conditional Statements When given several related conditional statements, you must put them into logical order. Example: Put the following statement in proper logical order. – Follow along with example in your textbook Look at the hypotheses and conclusions : The conclusion of one statement will flow into the hypothesis of the next. What do you look at in order to put conditional statements in logical order? See page 204 <ul><ul><ul><li>If Grant can play songs , then he can play in a band . </li></ul></ul></ul><ul><ul><ul><li>If Grant takes guitar lessons , then he can play chords . </li></ul></ul></ul><ul><ul><ul><li>If Grant can play in a band , then he can play a concert . </li></ul></ul></ul><ul><ul><ul><li>If Grant can play chords , then he can play songs . </li></ul></ul></ul><ul><ul><ul><li>If Grant buys a guitar , then he can take guitar lessons . </li></ul></ul></ul><ul><ul><ul><li>* You are looking for the conclusion (blue) of one statement flowing into the hypothesis (red) of another statement. </li></ul></ul></ul>Which conclusion does not flow into a hypothesis? If Grant can play in a band, then he can play a concert. The conclusion does not flow into a hypothesis so we know this is the last statement in logical order.
8. 8. LESSON 25: Logical Order of Conditional Statements <ul><ul><ul><li>If Grant can play songs , then he can play in a band . </li></ul></ul></ul><ul><ul><ul><li>If Grant takes guitar lessons , then he can play chords . </li></ul></ul></ul><ul><ul><ul><li>If Grant can play in a band , then he can play a concert . </li></ul></ul></ul><ul><ul><ul><li>If Grant can play chords , then he can play songs . </li></ul></ul></ul><ul><ul><ul><li>If Grant buys a guitar , then he can take guitar lessons . </li></ul></ul></ul>We know that A leads into C B leads into D D leads into A E leads into B C is the last one <ul><ul><ul><li>C. If Grant can play in a band , then he can play a concert . </li></ul></ul></ul>Which statement should go directly before C? <ul><ul><ul><li>A. If Grant can play songs , then he can play in a band . </li></ul></ul></ul>Which statement should go directly before A? <ul><ul><ul><li>D. If Grant can play chords , then he can play songs . </li></ul></ul></ul>Which statement should go directly before D? <ul><ul><ul><li>B. If Grant can take guitar lessons , then he can play chords . </li></ul></ul></ul>Which statement should go directly before B? <ul><ul><ul><li>D. If Grant buys a guitar , then he can take guitar lessons. </li></ul></ul></ul>
9. 9. <ul><li>LESSON 25: Logical Order of Conditional Statement </li></ul><ul><li>The statements in proper logical order: </li></ul><ul><li>E. If Grant buys a guitar, then he can take guitar lessons . </li></ul><ul><li> conclusion </li></ul><ul><li>B. If Grant takes guitar lessons , then he can play chords . </li></ul><ul><li>hypothesis conclusion </li></ul><ul><li>D. If Grant can play chords , then he can play songs . h y pothesis conclusion </li></ul><ul><ul><ul><li>A . If Grant can play songs , then he can play in a band . </li></ul></ul></ul><ul><ul><ul><li>hypothesis conclusion </li></ul></ul></ul><ul><li>C. If Grant can play in a band , then he can play a concert. </li></ul><ul><li>hypothesis </li></ul>
10. 10. LESSON 25: Critiquing for Validity Remember during our vocabulary review that Deductive reasoning can either be valid or invalid . We will now learn how to critique deductive arguments to determine if they are valid or invalid. Example: This is an example of valid reasoning: If a 4-sided polygon has 4 right angles, then it is a rectangle. Polygon ABCD has four right angles Therefore, polygon ABCD is a rectangle . Is the first statement true, assumed to be true or not necessarily true? The first statement is true by definition of a rectangle TRUE Is the second statement true, assumed to be true or not necessarily true? The second statement is assumed true because it is given ASSUMED TRUE Is the third statement true, assumed to be true or not necessarily true? The third statement is true because the first statement which is true tells us that a polygon with 4 right angles is a rectangle and Polygon ABCD has four right angles. TRUE
11. 11. LESSON 25: Critiquing for Validity Example: This is an example of invalid reasoning: Follow along on page 206 A person who lives in Arizona also lives in the United States. Mike lives in the United States. Therefore, Mike lives in Arizona. Is the first statement true, assumed to be true or not necessarily true? The first statement is true. Everybody who lives in Arizona also lives in the United States. TRUE Is the second statement true, assumed to be true or not necessarily true? The second statement is assumed to be true because Mike can live in the United States ASSUMED TRUE Is the third statement true, assumed to be true or not necessarily true? The third statement is not necessarily true because Mike can live in Florida. Florida is still in the United States NOT NECESSARILY TRUE
12. 12. LESSON 25: Converse, Inverse, Contrapositive Recall that the form of a conditional statement if p then q. You can make new conditional statements by switching p and q , negating both p and q or by switching and negating at the same time The converse f a conditional statement is formed by switching the places of the hypothesis and the conclusion. The sentence if p, then q becomes if q, then p What is a Converse or a conditional statement? See page 208 Example: If the phone rings then someone is calling you Converse If someone is calling you then the phone is ringing If a number is even then it is divisible by 2 Who can tell me the converse of this statement? If a number is divisible by 2 then it is even
13. 13. LESSON 25: Converse, Inverse, Contrapositive The converse of a true statement does not always produce a true statement or the converse of a false statement does not always produce a false statement. If the phone rings then someone is calling you If someone is calling you then the phone is ringing If a number is even then it is divisible by 2 If a number is divisible by 2 then it is even Is the converse of the statement true or false? The converse is false because your phone can be on silent. Is the converse of the statement true or false? The converse is true by definition of an even number.
14. 14. LESSON 25: Converse, Inverse, Contrapositive The inverse of a conditional statement is formed by negating the hypothesis and the conclusion. The sentence if p, then q becomes if not p then not q. The inverse is written as if ~ p then ~ q If someone is calling you then the phone is ringing How would you write the inverse of this statement? See page 210 If someone is not calling you then the phone is not ringing If a number is even then it is divisible by 2 If a number is not divisible by 2 then it is not even How would you write the inverse of this statement? See page 210 Just like the converse, a true statement can produce a false inverse and a false statement can produce a true inverse
15. 15. LESSON 25: Converse, Inverse, Contrapositive The contrapositive of a conditional statement is formed two steps. Step 1: Form the converse If someone is calling you then the phone is ringing How would you write the contrapositive of this statement? See page 211 If the phone is ringing then someone is calling you Step 1: Form the Converse Step 2: Form the inverse of the converse If the phone is not ringing then someone is not calling you Step 2: Form the Inverse of the Converse The contrapositive is different then the converse and inverse. A true statement will always produce a true contrapositive and a false statement will always produce a false contrapositive