2.3 Deductive Reasoning

Symbolic Notation Conditional Statements can be written using symbolic notation. Conditional Statements can be written using symbolic notation. p represents hypothesis p represents hypothesis q represents conclusion q represents conclusion

Example If the sun is out, then the weather is good. If the sun is out, then the weather is good. p is the hypothesis: the sun is out p is the hypothesis: the sun is out q is the conclusion: the weather is good q is the conclusion: the weather is good

Conditional Statement If p, then q If p, then q Symbolic Notation: p q Symbolic Notation: p q

Converse To form converse of an “If p, then q” statement, switch p and q. If q, then p. To form converse of an “If p, then q” statement, switch p and q. If q, then p. Symbolic Notation: q p Symbolic Notation: q p

Biconditional Statement Both conditional statement and converse are true! Both conditional statement and converse are true! p if and only if q p if and only if q Symbolic Notation: p q Symbolic Notation: p q Biconditional Statements are true forwards and backwards so arrow goes both directions. Biconditional Statements are true forwards and backwards so arrow goes both directions.

Negation Symbolic Notation: ~ Symbolic Notation: ~ Written before the letter Written before the letter

Inverse Negate the conditional statement Negate the conditional statement Symbolic Notation: ~p ~q Symbolic Notation: ~p ~q

Contrapositive ‘Negative flipflop’ ‘Negative flipflop’ Symbolic Notation: ~q ~p Symbolic Notation: ~q ~p

Example 1 Let p be “the value of x is -4” and let q be “the square of x is 16” 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. a. Write p q in words. If the value of x is -4, then the square of x is 16. If the value of x is -4, then the square of x is 16. b. Write q p in words. b. Write q p in words. If the square of x is 16, then the value of x is 4. If the square of x is 16, then the value of x is 4.

Example 1 c. Determine if p q is true. c. Determine if p q is true. Not true because the converse in not true. Not true because the converse in not true. d. Write ~p ~q in words. d. Write ~p ~q in words. If the value of x is not -4, then the square of x is not 16. If the value of x is not -4, then the square of x is not 16. e. Write ~q ~p in words. e. Write ~q ~p in words. If the square of x is not 16, then x is not -4 If the square of x is not 16, then x is not -4

Inductive Reasoning When we make conjectures based on observations. 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. Ex. For 3 weeks, the cafeteria served pizza on Wednesday. I conclude next Wed. the cafeteria will have pizza.

Deductive Reasoning Use facts, definitions and accepted properties in logical order to write a logical argument. 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. Ex. Dictionaries are useful books. Useful books are valuable. Therefore, dictionaries are valuable.

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. 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. Deductive Reasoning.

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. 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. Inductive Reasoning-based on observations.

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. 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 Deductive Reasoning

Law of Detachment If p q is a true conditional statement and p is true, then q is true. 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. Ex. If I pass the test, then I get an A in geometry. I passed the test. So I got an A in geometry.

Law of Syllogism If p q and q r are true conditional statements, then p r is true. 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. 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. Conclusion. I passed the test so I get a new car.

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. 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.

Law of Detachment or Law of Syllogism #1 1. If an angle is acute, then it is not obtuse. 1. If an angle is acute, then it is not obtuse. 2. <ABC is acute. 2. <ABC is acute. 3. <ABC is not obtuse. 3. <ABC is not obtuse. Valid by Law of Detachment Valid by Law of Detachment

Law of Detachment or Law of Syllogism #2 1. Right angles are congruent. 1. Right angles are congruent. 2. <A <B 2. <A <B 3. <A and <B are right angles. 3. <A and <B are right angles. Invalid. Statement 1 is not a conditional statement. Invalid. Statement 1 is not a conditional statement.

Law of Detachment or Law of Syllogism #3 1. If you save a penny, then you have earned a penny. 1. If you save a penny, then you have earned a penny. 2. Art saves a penny. 2. Art saves a penny. 3. Art has earned a penny. 3. Art has earned a penny. Valid by Law of Detachment. Valid by Law of Detachment.

Law of Detachment or Law of Syllogism #4 1. If you are a teenager, then you are always right. 1. If you are a teenager, then you are always right. 2. If you are always right, then people will listen to you. 2. If you are always right, then people will listen to you. 3. If you are a teenager, then people will listen to you. 3. If you are a teenager, then people will listen to you. Valid by Law of Syllogism. Valid by Law of Syllogism.

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. 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. 2. Pat received a speeding ticket. 3. Pat was driving 50 miles per hour in a school zone. 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. Invalid. Pat could have received a speeding ticket for speeding on the highway.

Law of Detachment or Law of Syllogism #6 1. If m<2=40°, then m<3=140° 1. If m<2=40°, then m<3=140° 2. If m<3=140°, then m<4=40° 2. If m<3=140°, then m<4=40° 3. If m<2=40°, then m<4=40° 3. If m<2=40°, then m<4=40° 4. Valid by Law of Syllogism 4. Valid by Law of Syllogism

Write a conclusion using the true statements. If no conclusion is possible, write no conclusion.

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. 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? Conclusion? Tim will go to the hospital. Tim will go to the hospital. Law of Syllogism Law of Syllogism

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. 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? Conclusion? Hank will buy a new car. Hank will buy a new car. Law of Syllogism Law of Syllogism

Example 3 If two planes intersect, then their intersection is a line. Plane A and plane B intersect. If two planes intersect, then their intersection is a line. Plane A and plane B intersect. Conclusion? Conclusion? Plane A and plane B intersect in a line. Plane A and plane B intersect in a line. Law of Detachment Law of Detachment

Example 4 If you cut class, then you will receive a detention. You cut class. If you cut class, then you will receive a detention. You cut class. Conclusion? Conclusion? You received a detention. You received a detention. Law of Detachment Law of Detachment

Example 5 If Jay doesn’t work hard, then he won’t start the game and will quit the team. Jay quit the team. If Jay doesn’t work hard, then he won’t start the game and will quit the team. Jay quit the team. Conclusion? Conclusion? No conclusion. We do not why he quit the team. No conclusion. We do not why he quit the team.