2.3 Deductive Reasoning Review NCSCOS: (2.01, 2.02)
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1 2.3 Deductive Reasoning Review NCSCOS: (2.01, 2.02)
2 Example If the sun is out, then the weather is good. p is the hypothesis: the sun is out q is the conclusion: the weather is good
3 Conditional Statement If p, then q Symbolic Notation: p q
4 Converse To form converse of an If p, then q statement, switch p and q. If q, then p. Symbolic Notation: q p
5 Biconditional Statement Both conditional statement and converse are true! p if and only if q Symbolic Notation: p q Biconditional Statements are true forwards and backwards so arrow goes both directions.
6 Negation Symbolic Notation: ~ Written before the letter
7 Inverse Negate the conditional statement Symbolic Notation: ~p ~q
8 Contrapositive Negative flipflop Symbolic Notation: ~q ~p
9 Example 1 Let p be the value of x is -4 and let q be the square of x is 16 a. Write p q in words. If the value of x is -4, then the square of x is 16. b. Write q p in words. If the square of x is 16, then the value of x is 4.
10 c. Determine if p Example 1 q is true. Not true because the converse in not true. d. Write ~p ~q in words. If the value of x is not -4, then the square of x is not 16. e. Write ~q ~p in words. If the square of x is not 16, then x is not -4
11 Inductive Reasoning When we make conjectures based on observations. Ex. For 3 weeks, the cafeteria served pizza on Wednesday. I conclude next Wed. the cafeteria will have pizza.
12 Deductive Reasoning Use facts, definitions and accepted properties in logical order to write a logical argument. Ex. Dictionaries are useful books. Useful books are valuable. Therefore, dictionaries are valuable.
13 Inductive or Deductive?? 1. Julie knows that Dell computers cost less than Gateway computers. Julie also knows that Gateway computers cost less than compaq computers. Julie reasons that Dell computers cost less than compaq computers. Deductive Reasoning.
14 2. Mike knows that Garrett is a sophomore and Kyle is a junior. All the juniors Mike knows are older than Garret. Mike reasons that Kyle is older than Garret based on these observations. Inductive Reasoning-based on observations.
15 3. If you live in Nevada and are between ages of 16 and 18, then you must take driver s education classes to get your license. Mark lives in Nevada, is 16, and has his driver s license. Therefore, Mark took driver s education classes. Deductive Reasoning
16 Law of Detachment If p q is a true conditional statement and p is true, then q is true. Ex. If I pass the test, then I get an A in geometry. I passed the test. So I got an A in geometry.
17 Law of Syllogism If p q and q r are true conditional statements, then p r is true. Ex. If I pass the test, then I get an A in geometry. If I get an A in geometry, then I get a new car. Conclusion. I passed the test so I get a new car.
18 Example 6 Determine if statement (3) follows from statement (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid.
19 Law of Detachment or Law of Syllogism #1 1. If an angle is acute, then it is not obtuse. 2. <ABC is acute. 3. <ABC is not obtuse. Valid by Law of Detachment
20 Law of Detachment or Law of Syllogism #2 1. Right angles are congruent. 2. <A <B 3. <A and <B are right angles. Invalid. Statement 1 is not a conditional statement.
21 Law of Detachment or Law of Syllogism #3 1. If you save a penny, then you have earned a penny. 2. Art saves a penny. 3. Art has earned a penny. Valid by Law of Detachment.
22 Law of Detachment or Law of Syllogism #4 1. If you are a teenager, then you are always right. 2. If you are always right, then people will listen to you. 3. If you are a teenager, then people will listen to you. Valid by Law of Syllogism.
23 Law of Detachment or Law of Syllogism #5 1. If you drive 50 miles per hour in a school zone, then you will get a speeding ticket. 2. Pat received a speeding ticket. 3. Pat was driving 50 miles per hour in a school zone. Invalid. Pat could have received a speeding ticket for speeding on the highway.
24 Law of Detachment or Law of Syllogism #6 1. If m<2=40, then m<3= If m<3=140, then m<4=40 3. If m<2=40, then m<4=40 4. Valid by Law of Syllogism
25 Write a conclusion using the true statements. If no conclusion is possible, write no conclusion.
26 Example 1 If Tim gets stung by a bee, then he will get very ill. If he gets very ill, then he will go to the hospital. Tim gets stung by a bee. Conclusion? Tim will go to the hospital. Law of Syllogism
27 Example 2 If Hank applies for the job, then he will be the new lifeguard at the pool. If he is the new lifeguard at the pool, then he will buy a new car. Hank applies for the job. Conclusion? Hank will buy a new car. Law of Syllogism
28 Example 3 If two planes intersect, then their intersection is a line. Plane A and plane B intersect. Conclusion? Plane A and plane B intersect in a line. Law of Detachment
29 Example 4 If you cut class, then you will receive a detention. You cut class. Conclusion? You received a detention. Law of Detachment
30 Example 5 If Jay doesn t work hard, then he won t start the game and will quit the team. Jay quit the team. Conclusion? No conclusion. We do not KNOW why he quit the team.
31 Deductive reasoning and real life Many people use deductive reasoning in their jobs. A physician diagnosing a patient's illness uses deductive reasoning. A carpenter uses deductive reasoning to determine what materials are needed at a work site.
32 Real-World Connection Auto Maintenance 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 is battery dead. What conclusion can she/he make? The mechanic can conclude that the car will not start.
33 Geography Use the Law of Detachment and the Law of Syllogism to draw conclusions from the following true statements. 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. You can use the first two statements and the Law of Syllogism to conclude: If a river is more than 4000 mi long, then it is the longest river in the world. With this, the fact that the Nile is 4132 mi long, and the Law of Detachment, you can also conclude: The Nile is the longest river in the world.
34 Most people are not detectives, but as a young woman, the English writer Agatha Christie ( ) correctly deduced that many people would like to be. In 1920 she published her first book, a detective novel entitled The Mysterious Affair at Styles in which she introduced the eccentric and ultra-logical Belgian detective Hercule Poirot. In this and in many subsequent novels, Poirot solves mysteries not with guns or car chases but with logical reasoning. Hercule Poirot as played by David Suchet,
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