LOGIC

1
Geometry Summer School 2007

• Lesson 1
• LOGIC

2
Statements

• What is a statement?
• Are all statements true?
• What does it mean to disprove a statement?

3
Counterexamples

• Definition of a counterexample
• Find a counterexample for the following
  statement
• Statement All the people in this room are from
  Loudoun County.

4
Counterexamples

• Find a counterexample for each of the following
  statements
• Statement The sum of two numbers is
  always greater than the larger number.
• Statement The value of x2 is always greater
  than the value of x.

5
Conditional Statements

• What is a conditional statement?
• What is a hypothesis? A conclusion?
• What is if-then form?

6
Conditional Statements

• Examples of conditional statements. (For
  each, circle the hypothesis and underline
  the conclusion.)
• If I get a 90, my grade is a B.
• If I practice, I will get good at sports.
• If I am in this class, I am registered to
  take summer school Geometry.
• If my pet is a Labrador, he is a dog.
• Can you think of any???

7
Conditional Statements

• Conditional statements are either True
  or False

8
Conditional Statements

• Determine whether the following conditional
  statements are true or false. State your
  reasoning.
• If it is raining, the sun is hidden behind the
  clouds.
• If my grade is an A, then I got a 95.

9
Conditional Statements

• Determine whether the following conditional
  statements are true or false. State your
  reasoning.
• If an animal is a cat, then it is a mammal.
• If you are a girl, then you loved to play with
  Barbie when you were a kid.

10
Conditional Statements

• Some conditional statements may need to be
  rewritten so that they are in if-then form.
• We will go out and play if it is sunny.
• All birds have feathers.
• Guitar players are musicians.

11
Symbolic Notation

• Conditional statements can be (and often are)
  represented with symbols.
• (The usefulness of this will be apparent after
  we go through a few more slides.)
• Example

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Symbolic Notation

• If p represents You are an
  equestrian and q represents You
  love horses, write the conditional statement
  represented by
• p ? q

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Inverse

• What does it mean to negate a statement?
• What is an inverse?
• Write the inverse of the following statement and
  state if it is true
• If you are 5 years old, you are young.

14
Inverse

• For each of the following conditional statements,
  first state whether it is true or not. Then,
  write the inverse and state whether the inverse
  is true.
• If a shape has four sides, it is a square.
• If x 6, then x2 36.

15
Converse

• What is the converse of a conditional statement?
• How can I remember which is converse and which is
  inverse?
• State the converse of the following statement
• If I am tired, then I need sleep.

16
Converse

• For each of the following conditional statements,
  first state whether it is true or not. Then,
  write the converse and state whether the converse
  is true.
• If you score a 91, you get a B.
• If an animal is a mammal, then it is a dolphin.

17
Notation for Converse Inverse

• Now we will need to use the symbol.
• Write the if-then, inverse, and converse
  statements AND their symbolic notations for the
  following statement.
• All whales live in the ocean.

18
Contrapositive

• What is the contrapositive of a conditional
  statement?
• How is it represented using symbolic notation?
• Example If you are touring the Eiffel
  Tower, then you are in France.

19
Contrapositive

• What are the contrapositives of the following
  conditional statements? State if the
  contrapositives are true.
• If you are a soccer player, then you are an
  athlete.
• If you are taking Geometry, then you finished
  Algebra 1.

20
Biconditional Statements

• What is a biconditional statement?
• Example An amiable person is friendly.

21
Law of Detachment

• What is the Law of Detachment?
• Use the Law of Detachment to make a valid
  conclusion from the following statements
• If you have graduated from high school, you
  passed Geometry.
• Miss Radtke has graduated from high school.

22
Law of Syllogism

• What is the Law of Syllogism?
• Use the Law of Syllogism to create a new
  conditional statement from the following
  statements
• If it snows, school will be cancelled.
• If school is cancelled, we will sleep in
  late.

23
Using the Laws

• Use the Law of Syllogism or the Law of Detachment
  to make a valid conclusion or statement for the
  examples below.
• Mary goes to the movies every Friday and Saturday
  night. Today is Friday.
• If Rick takes Biology this year, then Joe will be
  Ricks lab partner. If Joe is Ricks lab
  partner, then Rick will get an A in Biology.

24
Using the Laws

• State which law is demonstrated in the example
  below.
• If you get an A on your math test, you can go to
  the movies. If you go to the movies, you will
  see your favorite actor.
• If you get an A on your math test, you will see
  your favorite actor.