Mathematical Reasoning & Proof

MATHEMATICAL REASONING & PROOF Reasoning ❑Reasoning is the process of drawing conclusions or inferences from facts or premises. ❑ Logic and reasoning are associated with problem- solving and critical thinking. ❑ There are two types of reasoning: inductive and deductive.  Inductive re...

MATHEMATICAL REASONING & PROOF Reasoning ❑Reasoning is the process of drawing conclusions or inferences from facts or premises. ❑ Logic and reasoning are associated with problem- solving and critical thinking. ❑ There are two types of reasoning: inductive and deductive.  Inductive reasoning ❑ The logical process in which multiple premises, all believed to be true or found true most of the time combined to obtain a specific conclusion. ❑ Is a process of reaching a conclusion based on a series of observations. ❑ A conclusion may or may not be valid. ❑ It draws general principles from specific instances ❑Forms generalizations based on the examination of specific examples, instances or events Examples of inductive reasoning 1st Premise: The chair in the living room is red 2nd Premise The chair in the dining room is red 3rd Premise The chair in the bedroom is red CONCLUSION Therefore, all the chairs in the house are red.  1st Premise James is a grandfather 2nd Premise James is bald CONCLUSION Therefore, all grandfathers are bald  1st Premise Rico is a first-year college student 2nd Premise Rico is a male CONCLUSION Therefore, all first-year college students are males. Inductive Reasoning Predict the next integer in the sequence. a. 4, 8, 12, 16, ? b. 1, 1, 2, 3, 5, 8, ? 1.4 Logical Reasoning & Proof (Mathematics in the Modern World)  Deductive reasoning The process of reaching conclusion based on previously known facts. The conclusions are correct and valid. It draws specific conclusion from general principles or premises. Forms generalizations based on already established or agreed assumptions, laws, procedures, & laws of logical reasoning. EXAMPLES OF DEDUCTIVE REASONING 1st Premise All men ore mortal. 2nd Premise President Marcos is a man CONCLUSION Therefore, President Marcos is mortal  1st Premise All first-year college students in the new curriculum take Mathematics in the Modern World 2nd Premise Matilda is a first-year college student in the new curriculum. CONCLUSION Matilda takes Mathematics in the Modern World 1st Premise All Filipinos eat Rice 2nd Premise Ronnie is a Filipino CONCLUSION Therefore, Ronnie eats rice  1st Premise All numbers ending in O and 5 are divisible by 5. 2nd Premise The number 35 ends with a 5 CONCLUSION Therefore, 35 is divisible by 5 Proof (Definition) Cambridge Dictionary A piece of information that shows something exists or is true A piece or pieces of evidence Calculations that show how a solution to a math problem is reached 1.4 Logical Reasoning & Proof (Mathematics in the Modern World) Mathematical Proof (Definition) It is a collection of statements arranged logically that explains why a math statement is true. All math statements require a formal proof that demonstrates they are valid or are true. 1.4 Logical Reasoning & Proof (Mathematics in the Modern World) Mathematical System A mathematical system consists of Math objects, their definitions, and descriptions of relations among them Axioms, assumptions about objects that do not require a formal proof; it also includes rules of logical reasoning Theorems, these are statements that come as a result of logical reasoning and the assumptions about the objects in the mathematical system 1.4 Logical Reasoning & Proof (Mathematics in the Modern World)