INDUCTIVE & DEDUCTIVE REASONING. THE LOGIC OF MATH The logic we use in math can fall into one of two categories:

INDUCTIVE & DEDUCTIVE REASONING

THE LOGIC OF MATH

The logic we use in math can fall into one of two categories:

INDUCTIVE or DEDUCTIVE

WITH INDUCTIVE REASONING…..

You start with an OBSERVATION…

And make a CONJETURE (or hypothesis)about everything in that category.

Small,Specific

Observation

Big, General Conjecture

FOR EXAMPLE

You observe that:

Tony has red hair and a short temper.

Tommy has red hair and a short temper.

Timmy has red hair and a short temper.

Based on these observations, you make a conjecture:

All people with red hair have a short temper.

OR… “if you have red hair, you have a short temper”

Inductive reasoning is used quite often in the sciences…

For example:

“China's Rare River Dolphin Now Extinct, Experts Announce” December 14, 2006

The rare Chinese river dolphin has gone extinct, according to scientists who could not find a single one of the animals during a six-week search on China's Yangtze River.”

Or….”IF you are a Chinese River Dolphin, THEN you are dead.”

Sherlock Holmes often used inductive reasoning….

He would observe:

that a particular dog would bark at strangers…

and…

This dog barks at everybody except its owner.

…that the dog would not bark at its owner.

From that he conjectured:

Or…”IF you are not the owner of the dog, THEN the dog will bark at you.”

If you observe the number sequence:

1, 1, 2, 3…

What is your conjecture about the next number?

The trouble with inductive reasoning is that it can sometimes be WRONG.

Recall the red hair, short tempered example:

Tony has red hair and a short temper.

Tommy has red hair and a short temper.

Timmy has red hair and a short temper.

From these observations we could have also concluded that

All people whose first name begins with “T” have red hair and short tempers.

OR… “IF your first name begins with a T, THEN you have red hair and a short temper.”

BUT….

What about Tyler, who has brown hair is extremely patient?

Tyler disproves your conjecture, he is a counterexample.

'Extinct' dolphin spotted in Yangtze River(Xinhua)

Updated: 2007-08-29 19:49

WUHAN -- A white-flag dolphin has been seen in Yangtze River in east China, just days after a leading Chinese scientist said the

animal was likely extinct.

And you remember our extinct Chinese river dolphin?

Whoohoo, I’m alive!!Hey, but wait, where’s the rest of my species????

Small, specific

observations

BIG, GENERAL CONJECUTRE

Inductive Reasoning

But…

a counter example will kill a conjecture

DEDUCTIVE REASONING

It starts with a general proposition or RULE.

And you are led to a SPECIFIC CONCLUSION.

You couple that with a FACT

BIG, GENERAL

RULE

Small, specific

conclusion

Example:

General Rule: The “next number” in a sequence is the sum of the previous two numbers.

Fact: The first numbers in the sequence are 1, 1, 2, 3.

Specific Conclusion: The next number in the sequence is 5.

NOTE: This is the ONLY possible conclusion about the next number in the sequence.

Proofs by deduction are more powerful than proofs by induction.

With deduction…

if you know the general rule is correct

and

if you know the fact is correct

then the conclusion is inescapable.

Example (ala Sherlock Holmes)

This dog barks at everybody except its owner.

The dog did not bark at the murderer.

The murderer is the owner of the dog.

General Rule:

Fact:

Conclusion:

The difficulty with deductive reasoning is usually:

making sure the general rule is correct.

knowing the general rule; or

To SUM UP…

Deductive Reasoning

But…A sloppy rule will make the specific conclusion invalid.

BIG, GENERAL

RULE

Small, specific

conclusion

TALK ABOUT IT

At your tables, come up with instances in your life where you use inductive and deductive reasoning.