Geometry/Chapter 2/Lesson 4

Introduction
We will be going over inductive reasoning, conjectures, and counterexamples. Let's begin.

Inductive Reasoning
Inductive reasoning is used to make generalized decisions when you find a pattern in a specific case and then write a conjecture, an unproven statement based on observations. Inductive reasoning is used in predicting, forecasting and behavior--basically, it's more useful than you think. For example, if you see someone push you, you use inductive reasoning to assume that the person that pushed you is angry with you.

Why? During your lifetime, you've noticed that people that don't like each other and choose to get physical with each other push each other--thus, making the generalization, or conjecture, that a person that pushes another person does not like the person they are pushing. This conjecture can be proven false, as the push might have been an accident or a friendly/playful gesture. Thus, conjectures can be false.

Counterexamples
If a conjecture is false, then you have to explain why it is wrong. This is when counterexamples come in. A counterexample is an example that disproves a conjecture.

Let's take, for example, the following conjecture: When an elephant walks by, any human being around it becomes alert. We can provide several counterexamples--but we will choose one.
 * Counterexample: A dead person cannot become alert to an elephant walking by it.

Whenever you see a conjecture, ALWAYS look for a counterexample. Never go straight to "true"! Although this rule, there are conjectures that are most definitely right! Let's take this one, for example: A right angle is only 90 degrees. There is no counterexample to this because this conjecture is correct. All right angles are equal to 90 degrees... no matter what!