Applying Deductive Reasoning

Section 2.3

Essential Question

• How do you construct a logical argument?

Deductive reasoning

• Uses facts, accepted rules, postulates, theorems, and definitions.

• JUST THE FACTS!!!

• Inductive reasoning is based on observations.

Logic Laws

• Law of Detachment:– Direct argument (if, then)– If the hypothesis is true, then the

conclusion must be true.

• Law of Syllogism:– Chain argument (if (pq)…and (qr) …

then (pr)

Using the Law of Detachment

• Problems where you are asked to use the law of detachment to verify whether a statement is true will have:

– A conditional statement

– A given statement.

– Then you will make a conclusion.

Example• If two segments BC and XY have the same length ,

then they are congruent. • Given that you know that BC = XY.

• Conditional statement: If two segments BC and XY have the same length , then they are congruent.

• Given: You know that BC = XY

• The conclusion is that the segments are congruent

Another example

• If Mary goes to the movies, then it is Friday or Saturday night. Mary went to the movies.

• Conditional statement:

• Given:

• What do we conclude is true?

If Mary goes to the movies, then it is Friday

Mary went to the movies.

It is Friday or Saturday

Make your own Law of Detachment Problem

• You will need a – conditional statement– A given statement

Law of Syllogism

• These problems have 2 or more conditional statements that are combined to make a new conditional statement.

• You will be asked to decide what the conclusion or new conditional statement should be.

Example

• Hypothesis• If we follow the rules, then we will get our

work done.If we get our work done, then we will do well on

the test.• Conclusion• If follow the rules, then we will do well on the

test.

Read the hypothesis and make a conclusion

• If Rick takes chemistry this year, then Jesse will be Rick’s lab partner. If Jesse is Rick’s lab partner, then Rick will get an A in chemistry.

• If Rick takes chemistry this year, then Rick will get an A in chemistry.

Can you think of your own syllogism problem?

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Using Inductive or Deductive reasoning

• Inductive reasoning, make an observation and make a conjecture based on the observation.

• Deductive reasoning, use definitions and facts to draw/prove a conjecture.

Proving statements: Look for a pattern

• (2)(1) = 2• (2)(3) = 6• (2)(5) = 10• (2)(7) = 14• (2)(9) = 18• (2)(11) = 22• (2)(13) = 26• (2)(15) = 30

Make a conjectureAn odd number times an even number is even.

Prove it:Let 2n represent an even number GivenLet m be an odd number Given2n(m) = 2(n*m) Associative Property2(n*m) is a product of 2 and a number, so it is even.