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Chapter 1

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Chapter 1

Section 1.1

Deductive versus Inductive Reasoning

- There are many ways in which a person can reason. Some of them are better suited to doing some things than others. There are two ways to reason that are commonly used in mathematics:
- Deductive Reasoning
- Inductive Reasoning

Inductive Reasoning uses a series of examples or observations to generate a conclusion about something more general. The examples are sometimes referred to as a specific instance of a certain concept or idea.

Example of Inductive Reasoning

Lincoln is on a penny (Lincoln is a specific instance of a President)

Jefferson is on a nickel (A nickel is a specific instance of a coin)

Roosevelt is on a dime

Washington is on a quarter

Conclusion: All US minted coins have a President on them.

Inductive reasoning has both its advantages and disadvantages. Its big advantages is it gives a way to identify patterns. The disadvantage is that it is not always possible to test the pattern in all possible cases. The number of cases might be infinite or too big to be practical.

In order to be able to handle conclusions about very large (or sometimes infinite) collections of specific things we sometimes apply deductive reasoning.

Deductive reasoning uses a collection of general principles (sometimes called a hypothesis or premise) to draw a conclusion about something.

Example of Deductive Reasoning

Hypothesis:

Dr. Daquila voted in the last election.

Only people over 18 years old can vote.

Conclusion :

Dr. Daquila is over 18 years old.

- Many mathematical ideas are developed using a combination of both inductive and deductive reasoning. This becomes a two-step process.
- In the first step inductive reasoning is used as a “discovery” process when examples are used to see if a pattern can be recognized or identified.
- In the second step deductive reasoning verifies the pattern discovered in the first step is true under certain assumptions.
- In each of the following explain if inductive or deductive reasoning is being used.
- An employee has worn a blue shirt every Friday they have worked at a company. A person concludes they will wear a blue shirt this Friday to work.
- An employee at a company must wear a uniform that has a blue shirt. A person concludes they will wear a blue shirt this Friday to work.
- I am eating at a vegetarian restaurant. I conclude all the dishes on the menu are meatless.
- After reading the entire menu I conclude all of the dishes on the menu are not meatless
- The key is to determine if some specific examples are being used to draw the conclusion or some general principle.

inductive

deductive

deductive

inductive

This is called a Venn Diagram for a statement. In the diagram the outside box or rectangle is used to represent everything. Circles are used to represent general collections of things. Dots are used to represent specific items in a collection. There are 3 basic ways that categories of things can fit together.

All democrats are women.

Some democrats are women.

No democrats are women.

women

women

democrats

women

democrats

democrats

One circle inside another.

Circles overlap.

Circles do not touch.

What statement do each of the following Venn Diagrams depict?

democrats

women

George Bush

Laura Bush

Laura Bush is a democrat.

George Bush is not a woman.

Putting more than one statement in a diagram. (A previous example)

Hypothesis:

Dr. Daquila voted in the last election.

Only people over 18 years old vote.

Conclusion:

Dr. Daquila is over 18 years old.

people over 18

voters

Dr. Daquila

IF__THEN__ Statement Construction

Many categorical statements can be made using an if then sentence construction. For example if we have the statement: All tigers are cats.

This can be written as an If_then_ statement in the following way:

If it is a tiger then it is a cat.

cats

tigers

This is consistent with how we have been thinking of logical statements. In fact the parts of this statement even have the same names:

If it is a tiger then it is a cat.

The phrase “it is a tiger” is called the hypothesis. The phrase “it is a cat” is called the conclusion.

Can the following statement be deduced?

Hypothesis:

If you are cool then you sit in the back.

If you sit in the back then you can’t see.

Conclusion

If you are cool then you can’t see.

There is only one way this can be drawn! So it can be deduced and the argument is valid!

people who can’t see

people who sit in back

cool people

One method that can be used to determine if a statement can be deduced from a collection of statements that form a hypothesis we draw the Venn Diagram in all possible ways that the hypothesis would allow. If any of the ways we have drawn is inconsistent with the conclusion the statement can not be deduced.

Example

Hypothesis:

All football players are talented people.

Pittsburg Steelers are talented people.

Conclusion:

Pittsburg Steelers are football players. (CAN NOT BE DEDUCED and Invalid!)

talented people

talented people

talented people

football players

football players

Pittsburg Steelers

football players

Pittsburg Steelers

Pittsburg Steelers

Even though one of the ways is consistent with the conclusion there is at least one that is not so this statement can not be deduced and is not valid.