1.13: Inductive and Deductive Reasoning
Learning Objectives
- Use inductive reasoning to draw a conclusion from a series of examples.
- Use deductive reasoning to draw a conclusion from facts or accepted statements.
- Compare and contrast inductive and deductive reasoning.
Inductive Reasoning
One method of reasoning is called inductive reasoning. This means drawing conclusions based on examples.
- Inductive reasoning is based on _____________________________.
For example, a dot pattern is shown below. Look at the first three pictures. Do you see the pattern?
How many dots would there be in the bottom row of a fourth pattern?
There will be 4 dots. There is one more dot in the bottom row of each figure than in the previous figure. Also, the number of dots in the bottom row is the same as the figure number.
What would the total number of dots be in the bottom row if there were 6 patterns?
There would be a total of 21 dots. The rows would contain 1, 2, 3, 4, 5, and 6 dots. The total number of dots is \begin{align*}1 + 2 + 3 + 4 + 5 + 6 = 21\end{align*}.
This dot pattern is an example of _____________________________ reasoning because we are basing our conclusions about what comes next on the examples we have seen so far.
Now look at a pattern of points and line segments.
For two points, there is one line segment with those points as endpoints:
For three non-collinear points (points that do not lie on a single line), there are three line segments with those points as endpoints:
For four points, no three points being collinear, how many line segments with those points as endpoints are there?
The 6 segments are shown below:
For five points, no three points being collinear, how many line segments with those points as endpoints are there?
Answer: 10.
When we add a 5th point, there is a new segment from that point to each of the other four points. We can draw the four new dashed segments shown on the next page. Together with the 6 segments for the 4 points above, this makes \begin{align*}6 + 4 = 10\end{align*} segments.
5 points have 10 segments that connect them, as you see above.
Again, this is another problem where we used inductive reasoning: we used examples to make conclusions.
Inductive reasoning about patterns is a natural way to study new material. But there is a serious limitation to inductive reasoning: no matter how many examples we have, examples alone do not prove anything. To prove relationships, we will learn to use deductive reasoning, also known as logic.
- Deductive reasoning, also known as _____________________, proves relationships.
Can you think of a time in your life when you did something different from a pattern?
Have you ever changed the pattern of events in your life?
Deductive Reasoning
We all use logic—whether we call it that or not—in our daily lives. And as adults we use logic in our work as well as in making the many decisions a person makes every day.
Deductive reasoning begins with accepted facts or statements which we know are true. Then, we draw conclusions based on those facts.
- Deductive reasoning starts with accepted _______________________ which we know are _____________________.
Example 1
Suppose Bea makes the following statements, which are known to be true.
- If Central High School wins today, they will go to the regional tournament.
- Central High School does win today.
Common sense tells us that there is an obvious logical conclusion if these two statements are true: Central High School will go to the regional tournament.
Reading Check:
You are given two facts in the boxes below. Fill in the last box with the conclusion.
5 is the sum of an even and an odd number. (This is true, since \begin{align*}5 = 2 + 3\end{align*})
The diagrams on this page are example of _____________________________ reasoning because we start with facts that we have accepted to be true.
Suppose the following two statements are true:
If you love me let me know. If you don’t then let me go. (A country music classic. Lyrics by John Rostill.)
You don’t love me.
Fill in the boxes and find the logical conclusion:
You have now worked with both inductive and deductive reasoning. They are different but not opposites. In fact, they will work together as we study geometry and other mathematics.
Inductive reasoning means reasoning from examples.
You may look at a few examples, or many. Enough examples might make you suspect that a relationship is true always, or might even make you sure of this. But until you go beyond the inductive stage, you cannot be absolutely certain that it is always true.
- _______________________________ reasoning is reasoning from examples.
That’s where deductive reasoning enters and takes over. We have a suggestion arrived at inductively. We then apply rules of logic to prove, beyond any doubt, that the relationship is true always.
- ________________________________ reasoning uses accepted facts and logic to prove that something is true.
Inductive means “lead to.”
Inductive reasoning can “lead” you in the right direction to a conclusion, but you won’t know for certain that you have arrived at the correct conclusion.
Deductive means “lead from.”
(Think of the Spanish word “de,” which means “from.”) Because deductive reasoning is coming from facts, you know you have arrived at a true conclusion.
- Inductive means ___________________ to.
- Deductive means lead _________________.
Reading Check:
1. Fill in the blank:
Deductive reasoning uses ________________________________ to draw a conclusion.
2. Which type of reasoning, inductive, or deductive, is used to prove statements are true?
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