G8 Inductive and Deductive Reasoning Notes PDF

GOO GOO Motivation Activity: Spot the Pattern! Activity: "Predict the Next Number" 1. Sequence: 3, 6, 12, 24... What is the next shape in the sequence? Why Study Reasoning?  1. Enhances problem-solving skills.  2. Builds logical thinking and decision- making abilities.  3. Essential...

GOO GOO Motivation Activity: Spot the Pattern! Activity: "Predict the Next Number" 1. Sequence: 3, 6, 12, 24... What is the next shape in the sequence? Why Study Reasoning?  1. Enhances problem-solving skills.  2. Builds logical thinking and decision- making abilities.  3. Essential for understanding mathematical proofs and concepts.  4. Develops critical thinking applicable in real-life situations. Inductive vs. Deductive Reasoning What is Deductive Reasoning?  Definition:  - Deductive reasoning starts with a general principle and deduces specific conclusions.  Example:  - All triangles have three sides. This shape is a triangle. Therefore, it has three sides.  Strength:  - Conclusions are logically certain if premises are true. Comparison: Inductive vs Deductive  Inductive Reasoning:  - Observes patterns to form generalizations.  - May not always be accurate.  Deductive Reasoning:  - Uses general principles to deduce specific conclusions.  - Logically certain if premises are true.  Deductive vs. Inductive Reasoning The difference: inductive reasoning uses patterns to arrive at a conclusion (conjecture) deductive reasoning uses facts, rules, definitions or properties to arrive at a conclusion.  Inductive reasoning - Think of it like a We start with specifics and move to generalities Deductive reasoning – think of it like a We start with generalities and move to specifics. Inductive Reasoning, involves going from a series of specific cases to a general statement. The conclusion in an inductive argument is never guaranteed. Example: What is the next number in the sequence 6, 13, 20, 27,… There is more than one correct answer.  Here’s the sequence again 6, 13, 20, 27,…  Look at the difference of each term.  13 – 6 = 7, 20 – 13 = 7, 27 – 20 = 7  Thus the next term is 34, because 34 – 27 = 7.  However what if the sequence represents the dates. Then the next number could be 3 (31 days in a month).  The next number could be 4 (30 day month)  Or it could be 5 (29 day month – Feb. Leap year)  Or even 6 (28 day month – Feb.)  Examples of Inductive Reasoning Some examples 1) Every quiz has been easy. Therefore, the test will be easy. 2) The teacher used PowerPoint in the last few classes. Therefore, the teacher will use PowerPoint tomorrow. 3) Every fall there have been hurricanes in the tropics. Therefore, there will be hurricanes in the tropics this coming fall.  Deductive Reasoning  Deductive Reasoning – A type of logic in which one goes from a general statement to a specific instance.  The classic example All men are mortal. (major premise) Socrates is a man. (minor premise) Therefore, Socrates is mortal. (conclusion) The above is an example of a syllogism. Syllogism: An argument composed of two statements or premises (the major and minor premises), followed by a conclusion. For any given set of premises, if the conclusion is guaranteed, the arguments is said to be valid. If the conclusion is not guaranteed (at least one instance in which the conclusion does not follow), the argument is said to be invalid. BE CARFEUL, DO NOT CONFUSE TRUTH WITH VALIDITY! Examples: 1. All students eat pizza. Claire is a student at ASU. Therefore, Claire eats pizza. 2. All athletes work out in the gym. Barry Bonds is an athlete. Therefore, Barry Bonds works out in the gym. 3. All math teachers are over 7 feet tall. Mr. D. is a math teacher. Therefore, Mr. D is over 7 feet tall.  The argument is valid, but is certainly not true.  The above examples are of the form If p, then q. (major premise) x is p. (minor premise) Therefore, x is q. (conclusion) Deductive Reasoning An Example: The catalog states that all entering freshmen must take a mathematics placement test. You are an entering freshman. Conclusion: You will have to take a mathematics placement test. Inductive or Deductive Reasoning? Geometry example… What is the measure of angle x? x Triangle sum property - 60◦ the sum of the angles of any triangle is always 180 degrees. Therefore, angle x = 30° Inductive or Deductive Reasoning? Geometry example… What is the next shape in the sequence? 90% of humans are right handed. Joe is human, therefore Joe is right handed. DEDUCTIVE You are a good student. You get all A’s Therefore your friends must get all A’s too INDUCTIVE All oranges are fruits. All fruits grow on trees Therefore, all oranges grow on trees DEDUCTIVE Mikhail hails from Russia and Russians are tall, therefore Mikhail is tall DEDUCTIVE  No triangle is a square because all triangles have three sides and squares have four sides.  No triangle is a square because all triangles have three sides and squares have four sides. DEDUCTIVE  Since more than half of all automobile accidents involve drivers under twenty-five, it follows that drivers under twenty- five are probably a greater driving risk than those older than twenty- five. Smith missed work today. He must be ill because in the past he’s only missed work when he’s been ill.  The chances of rolling a five with a die are one in six. The presidential candidate that Maine selects usually indicates the one who’ll be elected. So, it’s safe to say, “As Maine goes, so goes the nation.” Every class I’ve taken so far has had an even male- female distribution in it. It’s obvious, then, that the student population of this college is evenly divided between males and females. If the president stands for re- election, he’ll surely be elected. Anyone who thinks the president won’t run again just doesn’t understand politics or political ambition. So, it’s clear who the next president will be: the present incumbent. Sandy was either present or she knew someone who was present. If she was present, then she knows more than she’s admitting. If she knew someone who was present, then she knows more than she’s admitting. Either way, Sandy knows more than she’s admitting. The chances that there are atmospheric conditions similar to earth’s elsewhere in the universe are very high. So, extraterrestrial life probably exists. This argument is valid because its premises logically entail its conclusion, and any argument whose premises logically entail its conclusion is valid. 1) Every quiz has been easy. Therefore, the test will be easy. 2) The teacher used PowerPoint in the last few classes. Therefore, the teacher will use PowerPoint tomorrow. 3) Every fall there have been hurricanes in the tropics. Therefore, there will be hurricanes in the tropics this coming fall. What is the next number in the sequence 6, 13, 20, 27,…There is more than one correct answer.  However what if the sequence represents the dates. Then the next number could be 3 (31 days in a month).  The next number could be 4 (30 day month)  Or it could be 5 (29 day month – Feb. Leap year)  Or even 6 (28 day month – Feb.) Mathematical Systems The four parts of a mathematical system are: 1. “Undefined” terms Terms such as point, line, and plane are classified as undefined because they do not fit into any set or category that has been previously determined. Definitions ( defined terms ) The four characteristics of a good definition are: a. It names the term being defined; b. It places the term into a set or category; c. It distinguishes itself from other terms in that category ( without providing unnecessary facts) d. It is reversible. If a triangle is isosceles, then it has two congruent sides. If a triangle has two congruent sides, then it is isosceles. DEFINITIONS: Principles 3. Postulates ( axioms, conjectures ) Postulate – a statement that is assumed to be true. 4. Theorems Theorem – a statement that follows logically from previous definitions and principles; a statement that can be proved to be true. Corollary – a theorem that follows from another theorem as a “by-product”; a theorem that is easily proved as the consequence of another theorem. Lemma – a theorem that is introduced and proved so that a later theorem can be proved (“helping theorem”).  Postulate 1: Through any two points, there is exactly one line. Postulate 2: The measure of any line segment is a unique positive number. The measure (or length) of AB is a positive number, AB.  Postulate 3: If X is a point of AB and A-X-B (X is between A and B), then AX + XB = AB Postulate 4: If two lines intersect, then they intersect in exactly one point  Postulate 5: Through any three noncollinear points, there is exactly one plane. Postulate 6: If two planes intersect, then their intersection is a line.+  Postulate 7: If two points lie in a plane, then the line joining them lies in that plane. Theorem 1.1: The midpoint of a line segment is unique. Example 1: State the postulate or theorem you would use to justify the statement made about each figure.