Clickers Bellwork Translate the following statement into a conditional statement Angles measuring less than 90o are acute angles Write the converse, inverse & contrapositive of the conditional statement you just wrote Determine the validity of each of the four statements you’ve created Can the statement “legally” be changed into a bi-conditional statement

Bellwork Solution

Analyze Deductive Reasoning Section 2.3

The Concept Deductive reasoning is one of other major components in the formation of logical arguments Today we’re going to see how deductive reasoning works and how we can apply it to mathematics

Definition Deductive Reasoning Process of making logical statements by using facts, definitions, accepted properties and the laws of logic. Axis of symmetry Vertex

What’s the difference Inductive Reasoning Deductive Reasoning Uses specific examples to make conjectures for the larger case Does not use any information outside of direct observation Deductive Reasoning Uses accepted facts and laws to make conjectures without the use of specific examples For example All the other cars around me are going 55 mph, I should too. The sign says the Speed Limit is 55 mph. Because it is an official sign placed by the proper authorities, I should follow it Other examples? Axis of symmetry Vertex

Reasoning from a graph Is the statement a product of inductive or deductive reasoning A 120 lb person uses twice as many calories when cycling 4 hours than when cycling 2 hours A. Inductive B. Deductive A 120 lb person uses fewer calories the longer he or she rides A. Inductive B. Deductive

Law of Detachment Law of Detachment If the hypothesis of a conditional statement is true, then the conclusion must also be true Basic assumption of truth based on credible conditional statements For example If two angles are right angles, then they are congruent. ABC & DEF are both right angles. If you are a UA student, then you are wearing a polo shirt. You are a UA student. Common parenting tool If you eat your chicken, then you’ll get dessert. You ate your chicken. Axis of symmetry Vertex

Example If x>12, then –x<-12. The value of x is 15. Axis of symmetry Vertex

Law of Syllogism Law of Syllogism Often associated with Socrates Process of stringing together two true conditional statements to prove a conditional statement If p, then q is true, and if q, then r is true. Then, if p then r is true Often associated with Socrates Used this method to both prove points and to disprove statements of others For example Xenophon, a tax collector, states that he is a pious person . 1: If you are a pious person, then you do not covet money 2. If you do not covet money, then you cannot be a tax collector 3. If you are a pious person, then you cannot be a tax collector Axis of symmetry Vertex

Examples What two conditional statements can we use to prove the following conditional statement If two angles are supplementary, then together they form a straight angle If two angles are supplementary, then form a 180o angle If an angle measures 1800, then it is a straight angle

Tandem The reason that both inductive a deductive reasoning are learned is so that conjectures can be made via experience and then proved via the laws of logic On your first day of school, you notice that everyone stays seated in a class until the teacher dismisses them Conjecture: In order to leave class at this school, you have to be dismissed by the teacher Proof: The teacher dictates how the students act The students act in a certain way when they leave a class The teacher determines how students leave a class

Examples What kind of statement can we make based on the following conditional using the law of detachment If the walls are painted red, then the students will get angry The walls just got painted red The students don’t like red The students will get angry The painters really like red Red is a fun color Vertex

Examples What kind of statement can we make based on the following conditionals using the Law of Syllogism If you aren’t wearing a belt, then you’re out of dress code If you’re out of dress code, then you have to visit the office If you out of dress code, then you’re not wearing a belt If you visit the office, then you aren’t wearing a belt If you visit the office, then you’re out of dress code If you aren’t wearing a belt, then you have to visit the office Vertex

Examples What kind of statement can we make based on the following conditionals using the Law of Syllogism If you don’t eat breakfast, then you’re metabolism slows down If you’re brain doesn’t function optimally, then you’re metabolism is slow If you don’t eat breakfast, then you’re brain doesn’t function optimally If you’re metabolism is slow, then you didn’t eat breakfast If you’re brain doesn’t function optimally, then you didn’t eat breakfast If you’re brain is slow, then your metabolism isn’t functioning optimally Vertex

Homework 2.3 1-3, 4-14 even, 16-19, 21-28

Examples What kind of statement can we make based on the following conditionals using the Law of Syllogism If you drive to school, then you park in the parking lot. If you have a parking permit, then you can park in the lot. Vertex

Most Important Points Deductive Reasoning Law of Detachment Law of Syllogism