Geometry Outline
1. Types of Reasoning
* Inductive Reasoning – * general conclusion based on a limited collection of specific observations * educated guesses * Primary flaw – we cannot be sure the conclusion is always correct * Counterexamples -- show a conclusion reached through inductive reasoning is not true
* Deductive Reasoning – * making a specific conclusion based on a collection of generally accepted assumptions. * There are no counterexamples * Premises – undefined terms, definitions and postulates (or previously proven theorems) * Fallacy: A conclusion that does not necessarily follow from the premises.
* Proof by Negation – Indirect Proofs – * Start with the Givens * Assume the negation of the Conclusion/Proof * The conclusion of the proof will be something that is wrong – contradicts another given. * So, what you want to prove must be true – the opposite of where you started. * Only works when what you want to prove has only two options – it either is or it isn’t. * Good when you are trying to prove that something is not true. Assume it is true and then prove that this makes no sense given what we are given.
2. Axiomatic System
* 4 Parts: * Undefined Terms: Starting point of a system. Have intuitive meaning. * Definitions: Statements that give meaning to new terms that will be used in a system. * Axioms or Postulates: Statements about undefined terms and definitions that are accepted as true without verification or proof. * Theorems: Statements that can be proven by using definitions, postulates and the rules of deduction and logic. * Corollary: a theorem that is easy to prove as a direct result of a previously proved theorem. * Conditional Statement: a statement that implies a cause and effect (if p, then q) * Hypothesis: The “If” Clause * Conclusion: the “Then” clause *
* Inductive Reasoning – * general conclusion based on a limited collection of specific observations * educated guesses * Primary flaw – we cannot be sure the conclusion is always correct * Counterexamples -- show a conclusion reached through inductive reasoning is not true
* Deductive Reasoning – * making a specific conclusion based on a collection of generally accepted assumptions. * There are no counterexamples * Premises – undefined terms, definitions and postulates (or previously proven theorems) * Fallacy: A conclusion that does not necessarily follow from the premises.
* Proof by Negation – Indirect Proofs – * Start with the Givens * Assume the negation of the Conclusion/Proof * The conclusion of the proof will be something that is wrong – contradicts another given. * So, what you want to prove must be true – the opposite of where you started. * Only works when what you want to prove has only two options – it either is or it isn’t. * Good when you are trying to prove that something is not true. Assume it is true and then prove that this makes no sense given what we are given.
2. Axiomatic System
* 4 Parts: * Undefined Terms: Starting point of a system. Have intuitive meaning. * Definitions: Statements that give meaning to new terms that will be used in a system. * Axioms or Postulates: Statements about undefined terms and definitions that are accepted as true without verification or proof. * Theorems: Statements that can be proven by using definitions, postulates and the rules of deduction and logic. * Corollary: a theorem that is easy to prove as a direct result of a previously proved theorem. * Conditional Statement: a statement that implies a cause and effect (if p, then q) * Hypothesis: The “If” Clause * Conclusion: the “Then” clause *