# Reasoning in Mathematics: Inductive and Deductive Reasoning

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Lesson Transcript
Instructor: Maria Airth

Maria has a Doctorate of Education and over 15 years of experience teaching psychology and math related courses at the university level.

Many people think that deductive and inductive reasoning are the same thing. It is assumed these words are synonymous. They are not. This lesson reveals the reality of these two types of reasoning.

## Deductive and Inductive Reasoning

You have probably heard the words inductive and deductive reasoning many times before. Chances are you have even mixed the two up, or think that they mean the same thing. But, they are not the same at all. In fact, they are actually opposites!

You can think of inductive and deductive reasoning as a path from something you know to something you don't know. Each makes use of general knowledge and knowledge of a specific scenario - think of it like the difference between a large lake and an individual fish living in the lake. The beginning point and ending points are switched on each.

This lesson will introduce each type of reasoning through definitions, general examples and mathematical examples. Join me as we learn to reason.

## Inductive Reasoning - Definition

Inductive reasoning starts with a specific scenario and makes conclusions about a general population. For our lake example, if you found a trout fish in a lake, you would assume that it is not the only fish in that lake. You may further conclude that all the fish in the lake are trout. You have induced from specific scenario information about the general population of the lake.

An interesting point with induction is that it allows for the conclusion to be false. It is simply a process of logical reasoning from a specific observation to a general theory of a population. The conclusion that all the fish in the lake are trout is very likely to be wrong; however, the process of induction was pure and logical. You could call it a valid guess.

## Examples of Inductive Reasoning

A great example of inductive reasoning is the process a child goes through when introduced to something new. If a child has a dog at home, she knows that dogs have fur, four legs and a tail. If a child were to be introduced to a cat, that child may very well assume the cat is a dog. Why? Because the cat has fur, four legs and a tail. In the child's experience, this means dog. The child induces from her specific scenario something about a much larger population.

Another example: You know two men from England who love soccer. You meet another man who states he is from England. You are shocked to find out he does not like soccer! Why were you surprised? You had induced from the two specific scenarios something about the entire population of English males.

Not all inductions are wrong though; but because they are broader, many can lead to inaccurate conclusions. Often, in the real world, this is seen as 'jumping to conclusions.'

Mathematically speaking, inductive reasoning might take this form:

Step 1 - show that something is true for a specific item.
Step 2 - show that if it is true for one, then it must be true for the rest.

Often these involve complicated mathematical proofs (something that is beyond the scope of this lesson), but a simple example is the induction that the sum of two odd numbers is even.

• Start with a specific true statement: 1 is odd and 3 is odd, the sum of which is 4; an even number.
• Now show it is true for the rest: an odd number is an even number plus 1. Thus two odd numbers are really two even numbers plus 2.
• The sum of even numbers is always even.

Starting with a specific observation, we were able to make an induction about the entire population of odd numbers. Here's an easy way to remember the direction of induction. 'In-' is the prefix for 'increase,' which means to get bigger. Thus, induction means start small and get bigger.

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