Now we will look at deductive reasoning, which uses logic to draw conclusions from given facts, definitions, and properties.

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1 Date: Lesson 2.3 Using Deductive Reasoning to Verify Conjectures Objective: Apply Law of Detachment and the Law of Syllogism in logical reasoning. TN Standard: Use the Law of Detachment, Law of Syllogism, conditional statements, and biconditional statements to draw conclusions. We have looked at specific cases and formed a conjecture based on those specific cases (inductive reasoning). Now we will look at deductive reasoning, which uses logic to draw conclusions from given facts, definitions, and properties. Sep 3 1:09 PM 1

2 Example of Inductive Reasoning or Deductive Reasoning? Andrea knows that Robin is a sophomore and Todd is a junior. All the other juniors that Andrea knows are older than Robin. Therefore, Andrea reasons that Todd is older than Robin. Inductive Example of Inductive Reasoning or Deductive Reasoning? Andrea knows that Todd is older than Chan. She also knows that Chan is older than Robin. Andrea reasons that Todd is older than Robin. Deductive Sep 3 1:27 PM 2

3 In the realm of deductive reasoning we have two math laws. These laws are used to help distinguish if statements are true or false. Sep 3 1:31 PM 3

4 The Law of Detachmentis one law of deductive reasoning. LAW OF DETACHMENT If p q is a true conditional statement and p is true, then q is true. Is the argument valid? Jamal knows that if he misses the practice the day before a game, then he will not be a starting player in the game. Jamal misses practice on Tuesday so he concludes that he will not be able to start in the game on Wednesday. p : If he misses practice the day before a game q : He will not be a starting player in the game p is true q is true Sep 3 1:41 PM 4

5 Is the argument valid? If two angles form a linear pair, then they are supplementary; A and B are supplementary. So, A and B form a linear pair. p: q: 2 angles form a linear pair they are supplementary q: is true p:? Sep 3 1:48 PM 5

6 The Law of Syllogism is another law of deductive reasoning. LAW OF SYLLOGISM If p q and q r are true conditional statements, then p r is true. Sep 3 1:49 PM 6

7 Determine if the conjecture is valid. Given: If m A < 90 0, then A is acute. If A is acute, then it is not a right angle. Conjecture: If m A < 90 0, then it is not a right angle. p: q: r: m A < 90 0 A is acute it is not a right angle. Sep 3 1:50 PM 7

8 Determine if the conjecture is valid. Given: If a number is divisible by 2, then it is even. If a number is even, then it is an integer. Conjecture: If a number is an integer, then it is divisible by 2. p: q: r: a number is divisible by 2 it is even it is an integer Sep 3 1:55 PM 8

9 How do we use this funny word? Write a conditional statement that can be made from the following true statements using the Law of Syllogism. 1. If a bird is the fastest bird on land, then it is the largest of all birds. 2. If a bird is the largest of all birds, then it is an ostrich. 3. If a bird is a bee hummingbird, then it is the smallest of all birds. 4. If a bird is the largest of all birds, then it is flightless. 5. If a bird is the smallest bird, then it has a nest the size of a walnut half shell. 1,2 3,5 1,4 Sep 3 1:49 PM 9

10 Can we use both laws at the same time? Over the summer, Mike visited Alabama. Given the following true statements, can you conclude that Mike visited the Civil Rights Memorial? If Mike visits Alabama, then he will spend a day in Montgomery. If Mike spends a day in Montgomery, then he will visit the Civil Rights Memorial. Sep 3 2:19 PM 10

11 Graphic Organizer of Lesson Copy the graphic organizer and give a definition or example for each based on the information from today's lesson. Types of Reasoning Inductive Reasoning Deductive Reasoning Law of Detachment Law of Syllogism Sep 3 1:57 PM 11

12 Assignment: Page 91 #'s 4 13, 17, 18, 20, 23 25, 27, 28 (18 problems) Sep 3 1:56 PM 12

13 Sep 3 2:03 PM 13