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Inductive and Deductive Reasoning
Objectives • Familiarize you with the deductive reasoning process • Learn the relationship between inductive and deductive reasoning
Problem Solving Logic – The science of correct reasoning. Reasoning – The drawing of inferences or conclusions from known or assumed facts. When solving a problem, • one must understand the question, • gather all pertinent facts, analyze the problem • i. e. compare with previous problems • (note similarities and differences), perhaps use pictures or formulas to solve the problem.
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.
Deductive reasoning is the process of reasoning logically from given statements to a conclusion.
Examples of Inductive Reasoning 1) Every quiz has been easy. Therefore, the test will be easy. 2) The teacher used Power. Point in the last few classes. Therefore, the teacher will use Power. Point tomorrow. 3) Every fall there have been hurricanes in the tropics. Therefore, there will be hurricanes in the tropics this coming fall.
Example of 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… x Triangle sum property – the sum of the angles of any triangle is always 180 degrees. Therefore, angle x = 30° 60◦ Deductive Reasoning – conclusion is based on a property
Inductive or Deductive Reasoning? Geometry example… What comes next? Is there a rule? Colored triangle rotating 90° CW in the corners of the square Inductive Reasoning
Who is known for using Deductive Reasoning?
Sherlock Holmes would use deductive reasoning to help solve crimes. Example: Sherlock Holmes could help solve a mystery by making inferences. If Holmes saw a pack of cigarettes by a victim (the victim did not smoke), Holmes can make the assumption that the killer is a smoker. Sherlock Holmes Andrew Ault
However… Deductive reasoning may not be the most accurate way of solving a problem, because we all know that assumptions can be wrong. Monty Python
How Can Deductive Reasoning Be Applied In School? Math Scott has a case of soda in his house since there are 13 cans of soda left I deduce that Scott has drank 11 cans of soda. English When I see ‘like’ in a sentence, I deduce that it is a simile.
Science Using laws and rules to make assumptions The law of gravity means everything that goes up must come down I threw a baseball in the air That means the baseball must come down. Social Studies To be elected President you must obtain at least 270 electoral votes George Bush won 287 electoral votes Therefore George Bush is the President.
What professions do you think commonly use Deductive Reasoning?
Why? These professions tend to ask a lot of questions to try to solve problems or to prove a point. Often they would have to make assumptions to solve problems. They would use rules and widely accepted beliefs to prove their argument. For example: An attorney states that his client is innocent because the crime victim was hit by a car. Since his client does not have a license. He can deduce that his client is innocent.
An auto mechanic knows that if a car has a dead battery, the car will not start. A mechanic begins work on a car and finds the battery is dead. What conclusion will she make? The mechanic can conclude that the car will not start.
Using Inductive Reasoning to Find a Pattern (yay Math!) 1, 2, 4, 5, 7, 8, 10, 11, 13, ___ Solution: the pattern seems to be to add 1, then add 2, then add 1, then add 2, … 0, 1, 1, 2, 3, 5, 8, 13, 21 ___, ___ Solution: It’s the Fibonacci Sequence, one of the most famous mathematical patterns! The pattern is after the 0 and the 1, you add the previous 2 numbers to get the next number. 0+1=1, 1+1=2 1+2=3…
Using Inductive Reasoning to Make a Conjecture (educated guess) Select a number: Add 50: Multiply by 2: Subtract the original number: Result: Now Prove it Using Deductive Reasoning!
Using Inductive Reasoning to Make a Conjecture (educated guess) Select a number: Add 50: Multiply by 2: Subtract the original number: Result: x x + 50 2(x+50) or 2 x + 100 - x X + 100
Deductive Reasoning This method of reasoning produces results that are certain within the logical system being developed. It involves reaching a conclusion by using a formal structure based on a set of undefined terms and a set of accepted unproved axioms or premises. The conclusions are said to be proved and are called theorems.
Deductive Reasoning 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!
Deductive Reasoning Examples: 1. All students eat pizza. Claire is a student at CSULB. 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.
Deductive Reasoning 3. All math teachers are over 7 feet tall. Miss Sweeney is a math teacher. Therefore, Miss Sweeney is over 7 feet tall. The argument is valid, but is certainly not true.
Examples No one who can afford health insurance is unemployed. All politicians can afford health insurance. Therefore, no politician is unemployed. VALID OR INVALID? ? ? The argument is valid
Finding a Counter Example to prove an argument wrong… Prove the statement is false: A number is divisible by 3 if it’s last two digits are divisble by 3. Counter Examples: 1527, 136, 418, and many more… The sum of any three odd numbers is even. Counter Examples: 1+1+1=3, 1+3+1=5, 1+3+7+11, and many more…