2.3 Deductive Reasoning

2.3 Deductive Reasoning Chapter 2: Reasoning and Proof Bellwork: pg 82 1-6

2.3 Deductive Reasoning Deductive Reasoning: Logical reasoning; reasoning from given statements to a conclusion Inductive Reasoning: Based on patterns you observe 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 will be needed at a work site.

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 battery is dead. What conclusion can she make? The mechanic can conclude that the car will not start.

Auto Maintenance Suppose that a mechanic begins work on a car and finds that the car will not start. Can the mechanic conclude that the car has a dead battery? No, there could be other things wrong with the car that would cause it not to start.

Law of Detachment If a conditional is true and its hypothesis is true, then its conclusion is true. Symbolic form: If p -> q is a true statement and p is true, then q is true.

Using the Law of Detachment For the given statements, what can you conclude? Given: If M is the midpoint of a segment, then it divides the segment into two congruent segments. M is the midpoint of AB Conclusion: M divides AB into two congruent segments, so AM = MB

Using the Law of Detachment 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? He should not pitch a complete game on Tuesday.

Real World Connection Does the following argument illustrate the Law of Detachment? Given: If it is snowing, then the temperature in less than or equal to 32°F. The temperature is 20°F. You conclude: It must be snowing. No, you can not make this conclusion.

Can you use the Law of Detachment? Given: If a road is icy, then driving conditions are hazardous. Driving conditions are hazardous. No conclusion can be made about whether or not the roads are icy.

Law of Syllogism Allows you to state a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of the other statements. Symbolic Form: If p -> q and q -> r are true statements, then p -> r is a true statement.

Using the Law of Syllogism 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. If a number is prime, then it is not a perfect square.

Using the Law of Syllogism to state a conclusion: 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. Conclusion: If a number ends in 0, then it is divisible by 5. If a number ends in 6, then it is divisible by 2. If a number ends in 4, then it is divisible by 2. No conclusion can be made.

Real World Connection 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. Conclusion: The Nile is the longest river in the world.

Real World Connection 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. Conclusion: The Volga River is not one of the world’s ten longest rivers and is less than 2300 mi long.

Homework • Pg 84 1-15

2.3 Deductive Reasoning