Inductive & Deductive Reasoning

MATH 102Contemporary Math

S. Rook

Overview

• Section 1.2 in the textbook:– Inductive reasoning– Deductive reasoning

Inductive Reasoning

Inductive Reasoning• Inductive Reasoning: the process of drawing a

conclusion by observing a pattern in specific instances. This conclusion is called a hypothesis or conjecture (guess)– i.e. going from specific examples to a general

conclusion– e.g. A stray animal comes by your house four

consecutive nights. You predict it will come on the fifth consecutive night

– e.g. You’ve always gotten a birthday card from your grandmother so naturally you expect to get a birthday card this year from your grandmother

Inductive Reasoning (Continued)

• The key to inductive reasoning is the observation of patterns:– e.g. 1, 2, 3, 4, ?, ?– e.g. -1⁄2, ¼, -1⁄8, ?, ?

• Inductive reasoning is based on pattern recognition, NOT scientific fact. Thus, it is possible for an inductive hypothesis to be incorrect– See example 4 on pg 21 in the textbook– e.g. The stray animal does not come on the 5th night– e.g. Your grandmother forgets to send you a birthday card

this year

Inductive Reasoning (Example)

Ex 1: Use inductive reasoning to solve.

a) 8, -4, 2, ?, ?, ?

b) The numbers 90, 264, and 8256 are all evenly divisible by 6. What can be noticed about all 3 numbers? Add the digits in each number, observe a pattern, and then form a conjecture

Deductive Reasoning

Deductive Reasoning• Deductive Reasoning: the process of using

accepted [scientific] facts and general principles to arrive at a specific conclusion– i.e. going from general (known) to the specific– e.g. A medical examiner uses his knowledge of the

temperature of the human body to determine the time of death of a murder victim

– e.g. Solving the equation 3x – 1 = 5 for x• Must feel comfortable in differentiating between

inductive and deductive reasoning– See exercises 7 – 16 on pg 23 of the textbook

Inductive & Deductive ReasoningEx 2: Consider the following process: (1) choose any

natural (positive) number (2) multiply the number by 3 (3) add 9 to the product you just found (4) divide the result by 3 (5) subtract the number you started with

a) Apply the process to three numbers. What are the results?b) Make a conjecture for part a). Does this involve inductive or deductive reasoning?c) Let n be the number you pick. Use the rules of Algebra to prove your conjecture. Does this involve inductive or deductive reasoning?

Summary

• After studying these slides, you should know how to do the following:– Reason inductively– Reason deductively– Understand the differences between inductive and

deductive reasoning• Additional Practice:– See the list of suggested problems for 1.2

• Next Lesson:– Estimation (Section 1.3)