Lesson 1: Mathematical Thinking and Logic
Objectives
By the end of this lesson, students will be able to: - decide whether information drawn from a passage is true, false, or cannot be determined
- recognize the hypothesis and conclusion of a conditional statement
- state conditionals in the
*if … then*form - recognize and state the converse, inverse, and contrapositive of a statement
- draw correct conclusions from given statements
Introduction Geometry
is a word taken from the Greeks with "geo" meaning
earth and "metron" meaning measure. In about 300 BC, a Greek
mathematician by the name of Euclid wrote a book
called In this lesson, you will apply critical thinking and decide whether
information drawn from a passage is true, false, or
cannot be determined. You will then learn how to
recognize the hypothesis and conclusion of a conditional statement and
how to state conditionals in the Critical Thinking The study of geometry requires one to be a critical thinker. Approaching and solving geometry problems is like a good detective analyzing a crime scene and looking for clues. You must be able to put your information in a clear, logical sequence to show how you arrived at your conclusion. Example 1 Read the sentence below and determine if the statements that follow are true, false, or cannot be determined.
As you can see from the example above, a valid conclusion cannot be drawn when you are missing a number of facts. Example 2 Read the following passage and determine if the statements that follow are true, false, or cannot be determined.
Practice Read the passage below and determine if the statements that follow are true, false, or cannot be determined.
Used with permission from David L. Williams 1. The rehearsal studio is located in New York City. Cannot be determined Check Your Answer
2. There were more than 160 actors that came to the audition. True Check Your Answer
3. Thomas was auditioning for a part. False Check Your Answer
4. Each actor selected would receive a paycheck. False Check Your Answer
5. Skye Benrexi will be selected for the part. Cannot be determined Check Your Answer
6. The rehearsal room is located at Harlequin Studios. True Check Your Answer
7. The rehearsal room was clean and fresh. False Check Your Answer
8. Skye auditioned well. True Check Your Answer
9. Thomas was a professional. True Check Your Answer
10. Skye Benrexi is a stage name. Cannot be determined Check Your Answer
Conditional Statements Look at the following conditional statement : If the Sun shines, then you will see your shadow. The first clause begins with the word In mathematics, we can label the if
Example 3 Identify the hypothesis and conclusion in the following statement: If I score a 93% on the test, then I will have an A average. hypothesis: I score a 93% on the test conclusion: I will have an A average Example 4 Identify the hypothesis and conclusion in the following statement: If the cows are lying down in the field, then it will rain. hypothesis: the cows are lying down in the field conclusion: it will rain Example 5 Identify the hypothesis and conclusion in the following statement: If hypothesis: conclusion: their product will be an integer Conditional statements are not always written in the Example 6 Identify the hypothesis and conclusion in the statement below.
Rewrite the statement so it is in the form When it rains, it pours. hypothesis: it rains conclusion: it pours
Example 7 Identify the hypothesis and conclusion in the statement below.
Rewrite the statement so it is in the form A banana is soft if it is spoiled. hypothesis: a banana is spoiled conclusion: it is soft
Practice 1-5. Identify the hypothesis and conclusion in the statements below. If a statement is not in the form if… then, rewrite the statement so it is in the form if… then. 1. If the lights are on in the house, then someone is home. hypothesis: the lights are on in the house conclusion: someone is home Check Your Answer
2. You should come inside if you get cold.
hypothesis: you get cold conclusion: you should come inside Check Your Answer
3. If a triangle has two equal sides, then it is an isosceles triangle. hypothesis: a triangle has two equal sides conclusion: it is an isosceles triangle Check Your Answer
4. The early bird gets the worm.
hypothesis: the bird is early conclusion: it gets the worm Check Your Answer
5. A straight angle measures 180 degrees.
hypothesis: an angle is a straight angle conclusion: it has a measure of 180 degrees Check Your Answer
Equivalent Statements The statement "A right angle has a measure of 90 degrees" can be written as a conditional statement: If an angle is a right angle, then it has a measure of 90 degrees. It is now in the form of "if If an angle has a measure of 90 degrees, then it is a right angle. When the If you live in Miami, then you live in Florida. Reversing the two parts of the statement gives us the converse: If you live in Florida, then you live in Miami. In this case the converse is not necessarily true. A person can live in other cities besides Miami and still be in Florida. If we want to negate or say the negative of a statement, we use the word If an angle is We use the symbol ~ for the word When you negate both the hypothesis and the conclusion, you have written the inverse. In the example above, both the conditional and its inverse are true, but this is not always the case. Look at the statement about Florida again: If you live in Miami, then you live in Florida. The inverse of this statement is not necessarily true: If you do not live in Miami, then you do not live in Florida. We can also take the inverse of the converse of a statement. Conditional statement: Converse: Take the inverse of the converse: If an angle does The inverse of a converse is called the contrapositive. For example, the contrapositive of the statement "If you live in Miami, then you live in Florida" is the following: If you do not live in Florida, then you do not live in Miami. In both cases above, note that the contrapositive is true. The contrapositive is always logically equivalent to the conditional statement (in other words, they both have the same meaning). Here is a way to remember conditionals: Example 8 Accept the following conditional statement as true: If you live in Winchester, then you attend St. Augustus High School. Write the converse, inverse, and contrapositive of each statement. Say whether each is true, false, or not necessarily true (cannot be determined), and which if any are logically equivalent.
Practice 1. Accept the following conditional statement as true: If you are on Flight #107, then you are going to Atlanta. Write the converse, inverse, and contrapositive of the statement, and say which statements, if any, are logically equivalent.
Check Your Answer
2. Identify the relation of each of the following statements to the original statement below: If you use a boogie board, then you live along the coast. a) If you do not live along the coast, then you do not use a boogie board. b) If you live along the coast, then you use a boogie board. c) If you do not use a boogie board, then you do not live along the coast. a) contrapositive b) converse c) inverse Check Your Answer
Deductive Reasoning In presenting a logical argument, one must combine at least three conditional statements in the following form (the symbol for "therefore" is ∴ ).
∴ Above, if the first two statements (premises) are true, then the third statement (conclusion) must also be true. Notice the hypothesis of one premise is the same as the conclusion of the other premise. This is called a syllogism. If the first and second premises are true, then the final conclusion is also true. Example 9 If a student walks to school, he is late for school. If a student is late for school, he misses part of his first class. What conclusion can you reach based on these premises?
Example 10 If you live in Tampa, then you live in Hillsborough County. If you live in Hillsborough County, then you live in Florida. What conclusion can you reach based on these premises?
Practice 1. Louis lives in the Ninth Ward. The Ninth Ward is in New Orleans. What conclusion can be reached from these premises? Therefore, Louis lives in New Orleans. Check Your Answer
2. Determine if the following pairs of premises form a syllogism. If they do, state the conclusion.
No (The hypothesis of one is not the conclusion of the other.) Check Your Answer
3. Determine if the following pairs of premises form a syllogism. If they do, state the conclusion.
Yes, Check Your Answer
Multiple Premises Sometimes there are more than two premises to consider.
∴ This is what a good detective does when solving a crime. One clue will lead to another. Eventually, when all the pieces fit together in the proper order, the crime is solved. Look at the premises below that lead to a conclusion. Notice the conclusion of one statement is the hypothesis of the next statement. If Alvin has $80, then he will go shopping. If Alvin goes shopping, then he will buy baseball shoes. If Alvin buys baseball shoes, then he will try out for the team If Alvin tries out for the team, then he will make the team. Therefore, if Alvin has $80, then he will make the baseball team. Sometimes the steps to prove a syllogism are not as obvious as those in the preceding problems. An equivalent statement may need to be written in order to have the conclusion of one premise be the hypothesis of the other. Equivalent statements can be substituted for one another. Recall that the conditional and its contrapositive are equivalent. Look at the examples below of equivalent statements:
~
Each of these could be substituted to produce a syllogism that leads to a conclusion. Example 11 Determine if the following pairs of premises form a syllogism. If they do, state the conclusion.
Use the fact that the contrapositive of Rewrite the pair of premises, changing the order and substituting the contrapositive, to find the conclusion.
∴ Practice 1. If Peter makes an A in his geometry class, he will spend the summer in the Caribbean. If he goes to the Caribbean, he gets to go to the summer festival. If he goes to the summer festival, he will perform with the band. What conclusion can be drawn from these premises? Check Your Answer
Truth Tables Earlier in the lesson you learned how conditional, converse, inverse, and contrapositive statements can be equivalent. Now we are going to verify which ones are equivalent by using a truth table. A truth table shows all the possibilities in a given situation. It is raining. You get wet. Both of these statements can either be true or false. Assign the letter If it is raining, then you get wet. In symbolic form this is written:
A truth table is a way to display all the possibilities about whether either of those statements are true. It shows the original, converse, inverse, and contrapositive. First look at a truth table for the two parts of the conditional statement above:
Notice that there are four possible combinations of true and false for
In the first case, T → T. If both of the statements--that it is raining and that you get wet --are true statements, then the conditional statement is true. So, the truth value of this is T. The second case is T → F. This means that the first statement, it is raining, is true, but the second statement, that you get wet, is not. If this is the case, the conditional statement cannot be true. The truth value here is F, false. In the third case, F → T. Here, the first statement is not true, so it is not raining. The second statement, you get wet, is true. In this case, the truth value is T, because we do not have enough information to know that it is false. In the fourth case, F → F. In this case, the first and second statements are both false. As with the example above, since we do not have any information about what happens when it is not raining, the statement cannot be proven false. Here, the truth value is T. Negation The symbol for negation (or "not") is a tilde (~). If we want to show that a statement
Negation causes a true statement to become false and a false one to become true. Say the original statement is "10 plus 5 equals 15." This is true. The negation of this statement is "10 plus 5 does not equal 15." This is false. In another example, the original statement is "A pig is a horse." This is false. The negation of this statement is "A pig is not a horse." This is true. Conjunction Another symbol that we use in truth tables is the conjunction or
"and" symbol. If we want to connect the statements Look back at the example above. We will connect the statements with a conjunction: It is raining and you get wet.
The truth table for conjunctions is the following:
Look at an example. Check to see if the conjunction statement is true based on the truth table: Green is a color Look at the table. Both of these statements are true. Based on the first row of the table, the entire sentence is true. Now look at a similar statement: Green is a color Since the first statement is true and the second is false, the entire sentence is false. Example 12 Use the truth table above to state the truth value of the sentence below. An apple is a vegetable The first statement is false--an apple is not a vegetable. The second statement is true. Since the first statement is false and the second is true, the entire sentence is false. Example 13 Use the truth table above to state the truth value of the sentence below. An apple is a vegetable The first statement is false. The second statement is also false. Since the first statement is false and the second is false, the entire sentence is false. Notice that the only time Practice 1 - 4. Determine the truth value of the following sentences. 1. 5 + 7 = 12 and 5 is an even number. The first statement is true, but second statement is false. Therefore, the whole sentence is false. Check Your Answer
2. London is the capital of England and Paris is the capital of France. Check Your Answer
3. Eleven is not a prime number and there are five sides in a pentagon. Check Your Answer
4. The opposite sides of a trapezoid are always equal and there are 350° in a circle. The first statement is false and the second statement is false, so the whole statement is false. Check Your Answer
Equivalent Statements Look again at the statements we have been using:
We can use this statement and truth tables to show which statements are equivalent.
If it is raining, then you get wet.
We have seen this truth table before.
If you get wet, then it is raining.
Notice that the
We have labeled the columns with Roman numerals. In order to get column III, you must do column II → column I. Notice that the results are different from A.
If it is not raining, then you do not get wet. ~ Because
If you do not get wet, then it is not raining. ~ Notice that the
Notice that the last column in the conditional (A) and the contrapositive (D) tables are the same. Therefore, we say that they are equivalent statements. Also, the last column in the converse (B) and the inverse (C) tables are the same. Therefore, they are equivalent. Validity of an Argument Truth tables can also be used to determine the validity of an argument. An argument consists of premises (hypothesis) and a conclusion. The premises are connected with the word "and" (shown with the symbol ∧). Look at an example of an argument.
Write the statements with symbols.
Put premises together with the symbol for a conjunction (∧). ( The conclusion is [( Let's build a truth table to look at this statement.
Notice that we start with
Column III is formed by using the rules for → (implied). III = I → II Column IV is formed by using the rules for ∧ (and). IV = III ∧ I Column V is formed by using the rules for → (implied). V = IV → II In the last column (V), the results are all true. When this happens it is called a tautology, and the argument is valid. Look at another argument. If it is raining, then I am reading. I am reading. Therefore, it is raining.
Now set up the truth table:
Column III is formed by using the rules for → (implied). III = I → II Column IV is formed by using the rules for ∧ (and). IV = III ∧ II Column V is formed by using the rules for → (implied). V = IV → I In the last column (V), the results are not all true. Therefore, the argument is not valid. Here is another argument. Example 14 Construct and describe a truth table about the following argument. If wool is expensive, then sweaters are expensive. Sweaters are not expensive. Therefore, wool is not expensive.
Notice that there are negations needed here. Add negation columns to the truth table.
Columns III and IV are formed by negating Columns I and II. Column V is formed by using the rules for → (implied). V = I → II Column VI is formed by using the rules for ∧ (and).VI = V ∧ IV Column VII is formed by using the rules for → (implied). VII = VI → III The last column is all true. Therefore, the argument is valid. Practice 1. Construct and explain a truth table about the following argument. If John does not go to town, then Marty stays home. John goes to town. Therefore, Marty does not stay home.
Columns III and IV are formed by negating Columns I and II Column V is formed by using the rules for → (implied). V = III → II Column VI is formed by using the rules for ∧ (and). VI = V ∧ I Column VII is formed by using the rules for → (implied). VII = VI → IV The last column is not all true. Therefore, the argument is not valid. Check Your Answer
Enrichment Activity Research the history of geometry. There are many resources that
contain information on how geometry was first used by
the Babylonians, Egyptians, Hindus, Chinese and
Greeks. Famous mathematicians such as Euclid, Archimedes, Thales, and
Pythagoras were instrumental in the development of
geometry. Do some reading on these ancient times and
mathematicians, and write a summary of what you have learned. One
possible resource is a book titled Lesson Review & Homework We started this lesson by defining conditional statements, also known as You used this information when applying deductive reasoning and finding syllogisms, which are used to form a true conclusion from given premises. You learned about truth tables. You also learned how to use them to understand negation, conjunctions, and the validity of arguments. This kind of logical thinking will be the basis for much of the work you will do as you explore geometry. Homework 1. Identify the hypothesis and conclusion of each of the following
conditional statements. Rewrite any statement
that is not already in the a. If the geese are flying south, then winter is on its way. b. If you study hard, then you will achieve success. c. The diameter of a circle is 14 inches if the radius of the circle is 7 inches. d. Those that live in Florida do not have to pay a state income tax. e. A scalene triangle has no sides of equal length. 2. Identify the relation of the following statements to the original statement below: The grass will grow if it rains. a. If it doesn't rain, the grass will not grow. b. If the grass grows, then it rained. 3. Write the converse of the statement below. Is it a good definition of what it means for an animal to be extinct? Why or why not? If a type of animal is extinct, then there are no animals of that type left on Earth. 4. Determine if the following pairs of premises form a syllogism. If they do, state the conclusion.
5. Determine if the following pairs of premises form a syllogism. If they do, state the conclusion.
6. Determine if the following pairs of premises form a syllogism. If they do, state the conclusion.
7. Determine if the following pairs of premises form a syllogism. If they do, state the conclusion ~
8. Form the negation of each statement. a. The sky is cloudy. b. The Pittsburgh Steelers football team is not the team with the most Super Bowl wins. c. Josie is the only person with red hair. 9. Let
a. If this is a cow, then this is a mammal. b. If this is a mammal, then this is a cow. c. If this is not a mammal, then this is not a cow. 10. Let
a. b. c. ~ d. ~ 11. 11. Use truth tables to show that 12. By using a truth table, determine the validity of the following argument: If we turn off the TV, there will be less noise. There is less noise. Therefore, we turned off the TV. |