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Explain in your own words what deductive reasoning is.

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### Deductive Reasoning II

“Education is the most powerful weapon which you can use to change the world.” ― Nelson MandelaDo Now

Explain in your own words what deductive reasoning is.

Exit Slip Error Analysis

Compare your exit slip to this one.

Judging the validity of conditional statements

Today’s Objectives

Explain the laws used in the deductive reasoning process.

Use deductive reasoning to lead to accurate conclusions.

Use the Law of Detachment

Use the Law of Syllogism

Use Problem Solving Skills

Rewrite in conditional (“if-then”) form

All quadrilaterals have four sides.

If it’s a quadrilateral, then it has four sides.

Inverse?

If it’s not a quadrilateral, then it does not have four sides.

In other words, it has more or less than four sides.

Rewrite in conditional (“if-then”) form

A triangle has, at most, one right angle.

If it’s a triangle, then it has, at most, one right angle.

Inverse?

If it’s not a triangle, then it has more than one right angle.

Rewrite in conditional (“if-then”) form

Two lines in a plane always intersect at exactly one point

If there are two lines in a plane, they intersect at exactly one point

Negation?

If there are two lines in a plane, they do not intersect at exactly one point.

In other words, they do not intersect at all or they intersect at more than one point.

Counterexample: Parallel lines!

Some terms

Axiom – a self-evident truth that requires no proof; a statement accepted as fact

Postulate – a proposition that requires no proof

Theorem – a proposition that can be deduced from the premises or assumptions of a system

Corollary – a proposition that is incidentally proved in proving another proposition

Reflexive Property

A=A

A quantity is equal to itself

In logic, A A.

Always true in logic

If you’re a student at Simon Tech, then you’re a student at Simon Tech.

If a pentagon has five sides, then a pentagon has five sides.

Symmetric Property

If A=B then B=A

Always true of numbers (if x=5 then 5=x)

In logic, If A B, then B A.

Not always true.

If I eat too much I get sick. If I get sick then I eat too much.

But when is it true?

When the Biconditional statement is true.

For example, “Two lines intersect iff they are not parallel”

Transitive Property

- Also known as the Law of Syllogism
- If A=B and B=C then A=C
- In logic, If A B and B C, then A C.
- For example:
- If the electric power is cut, then the refrigerator does not work.
- If the refrigerator does not work, then the food is spoiled.
- So if the electric power is cut, then the food is spoiled.

Law of Detachment

- Also known as Modus Ponens
- If P Qis true and P is true, then Q must be true.
- For example
- If an angle is obtuse, then it cannot be acute.
- Angle A is obtuse.
- Therefore, Angle A cannot be acute.

Law of Syllogism

Transitive Property

Is this valid?

Christian wrote the following argument:

If the soccer team loses, Janalee won’t watch their next game.

Janalee watched their next game.

Therefore, the soccer team lost.

No. But what can Christian conclude?

The soccer team won.

Is this valid?

Jessica wrote the following argument:

If the sun is out, then Karina will go to the beach.

If she does not go with friends, then Karina will not go to the beach.

The sun is out.

Therefore, Karina goes with friends.

Today’s Objectives

Explain the laws used in the deductive reasoning process.

Use deductive reasoning to lead to accurate conclusions.

Use the Law of Detachment

Use the Law of Syllogism

Use Problem Solving Skills

Exit Slip

For #1, fill in both blanks and explain your reasoning.

- Using the Law of ____________, what can be deduced? If you check your email, you must have internet access. Michael checks his email. Therefore, _______________________________.
- Describe the following properties:
- Reflexive
- Symmetric
- Transitive
- Five girls took part in a race. Ana finished before Blanca but behind Concepcion. Daysi finished before Elizabeth but behind Blanca. What was the finishing order?
- Explain your reasoning for #3. Include whichlaw or property you used.

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