Calculus Fall 2010 Lesson 01



Lesson Plan #007

Class: Geometry Date: Friday/Monday October 2nd/5th

Topic: Connectives in logic Aim: What are some connectives in logic?

HW #007:

HW #7 is on page 5 of this lesson plan

Objectives:

1) Students will be able to use connectives in logic.

Do Now:

1) Which point is closest to the orthocenter of the triangle?

PROCEDURE:

Write the Aim and Do Now

Get students working!

Take attendance

Give Back HW

Collect HW

Go over the Do Now

Up to this point we have worked with a few definitions. We would like to start using those definitions to prove theorems. For example, here is one of my favorite theorems that I would like to prove so I could use it already.

Theorem: If two angles are vertical angles, then they are congruent.

Use the above theorem for vertical angles to solve for x. 

Before we proceed using this theorem or any other theorems, we would like to prove those theorems. To prove theorems, we have to learn the basic laws of reasoning. This study of reasoning is called logic.

To start let’s determine if the following sentences are true?

1) Shrimp tastes good!

2) 3x + 2 = 7

3) He is my friend.

The first sentence has an uncertain truth value since it is true for some people but not others.

The 2nd and 3rd sentences are open sentences, since they contain variables and we can’t assign a truth value to the sentence until we choose a replacement for the variable.

Determine the truth value of the sentences below:

1) The degree measure of a right angle is 90.

2) 3(3) + 2 =17

3) One is a prime number

A sentence that can be judged to be either true or false is called a statement or closed sentence. A statement contains no variables.

In logic, we study the truth value of statements; some of the statements are simple and some are compound statements, or more than one statement, connected by connectives such as and, or, if…then, … if and only if….

Determine whether the following statement is true or false:

The altitudes of a triangle are concurrent.

How would you write the negation of this statement?

The negation of a statement is usually formed by placing the word not within the original or given statement.

To show the negation of a statement in symbolic form, we place the symbol ~ before the letter.

Is the negation of the original above statement true or false?

A statement and its negation have opposite truth values.

A truth table is a compact way of listing symbols to show possible truth values for a set of statements.

1. Which of the following is a closed sentence?

A) Summer follows spring. B) A quarter is a coin. C) There are 360 days in a year.

D) All of the above.

2. What is the negation of, "Jenny rides the bus"?

A) Jenny does not like to ride the bus. B) Jenny does not ride the bus.

C) Jenny likes to ride the bus. D) None of the above.

3. Which of the following is the negation of x?

A) –x B) ~(~x) C) ~x D) None of the above.

4. Given:

a: A triangle is not a polygon.

b: A square is a rectangle.

Problem: Which of the following is the negation of "A triangle is not a polygon"?

A) ~(~b) B) ~a C) a D) None of the above.

5. Which of the following is an open sentence?

A) The number 4 is even. B) The number 8 is odd. C) The number 5 is even. D) The number x is odd.

6)

A conjunction is a compound statement formed by joining two statements with the connector AND. The conjunction "p and q" is symbolized by p[pic]q. A conjunction is true when both of its combined parts are true; otherwise it is false.

|Example 1: |

|Given: |

|p: Ann is on the softball team. |

| |

| |

|q: Paul is on the football team. |

| |

|Problem: |

|What does p[pic]q represent? |

| |

Complete the table at right:

Complete the truth tables below:

|[pic]   |

Exercises:

1. Which of the following sentences is a conjunction?

A) Jill eats pizza or Sam eats pretzels. B) Jill eats pizza but not pretzels.

C) Jill eats pizza and Sam eats pretzels. D) None of the above.

2. Which of the following statements is a conjunction?

A) p + q B) p ^ q C) ~p D) None of the above.

3. A conjunction is used with which connector?

A) Not B) Or C) And D) None of the above.

4. If a is false and b is true, what is the truth value of a ^ b?

A) True B) False C) Not enough information was given. D) None of the above.

5.Given:

r: y is prime.

s: y is even.

Problem: What is the truth value of r^s when y is replaced by 2?

A) True B) False C) Not enough information was given. D) None of the above.

A disjunction is a compound statement formed by joining two statements with the connector OR. The disjunction "p or q" is symbolized by p[pic]q. A disjunction is false if and only if both statements are false; otherwise it is true.

|Given: |p: Ann is on the softball team. |

| |q: Paul is on the football team. |

|Problem: |What does p[pic]q represent? |

Complete the table at right:

Complete the truth tables below

[pic]

Exercises:

1. Which of the following sentences is a disjunction?

A) Amy played soccer or Bill played hockey. B) Amy played soccer and Bill played hockey.

C) Amy did not play soccer and Bill played hockey. D) None of the above.

2. Which of the following statements is a disjunction?

A) ~x ^ y B) x ^ y C) x[pic]y D) None of the above.

3. A disjunction is used with which connector?

A) And B) Or C) Not D) None of the above.

4. If a is false and b is true, what is the truth value of a[pic]~b?

A) True B) False C) Not enough information was given D) None of the above.

5. Given:

r: y is prime.

s: y is even.

Problem: Which of the following is a true statement when y is replaced by 3?

A) r[pic]~s B) r[pic]~s C) r[pic]s D) All of the above.

Geometry HW#7:

Name: _____________________________ Date ___________ Period ______________

1) What is logic?

2) What is a statement in logic?

3) Give an example of a true statement.

4) What is the purpose of a truth table?

5) When will a conjunction statement be true?

6) When is a disjunction false?

7)

8)

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