Lesson 4

Lesson 4: Identifying Errors in Proofs

 

Focus

 

 

Many types of puzzles or problems involve the use of reasoning to solve a mystery. TV shows like CSI or NCIS often have the lead detectives make a conjecture about the identity of the murderer and then justify their reasoning through investigation.

 

Not all reasoning used to justify a proof is valid. For instance, circumstantial evidence often points at one suspect and, often times, once the forensic analysis is done, a different suspect is identified.

 

This lesson will help you answer the following inquiry question:

• How can the validity of a proof be justified?

Assessment

• Lesson 4 Assignment

All assessment items you encounter need to be placed in your course folder.

 

Save a copy of the Lesson 4 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

 

Discover

 

In order for a proof to be valid, the arguments used to support the generalization or conclusion must be valid.

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Try This 1

 

Consider the following sets of integers:

 

10, 11, 12

5, 6, 7, 8, 9

1, 2, 3, 4, 5, 6, 7

1. Show that the sum of each set of consecutive integers divided by the number of integers in the set is equal to the middle term in each set of numbers. For example,
   
   
   
   

2. Form a deductive proof that the sum of any set of 5 consecutive integers divided by the number of integers in the set (5) is equal to the middle term in each set.

Share 1

 

Having a discussion with a partner or group will help to identify alternative strategies that may be effective in developing a deductive proof. As well, you may discover a way to revise your personal strategy to make it more efficient.

 

Compare your answers from Try This 1 with a partner or in a group. In your discussion, discuss the following questions:

• When you divided the sum of each set of numbers by the number of integers in the set in question 1, did you get the middle term of the set?

• What strategy did you each use to write a deductive proof for question 2? What did your proof look like?

• Did the strategies used result in a valid proof? If not, can you identify any errors in the proof?
  

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After completing the preceding activities, Joel decides that he wants to look at an even number of consecutive integers. He decides to prove that the sum of 4 consecutive integers is evenly divisible by the number of integers in the set (4).

 

You can watch the animation titled Joel’s Solution.

 

 

A copy of Joel’s Solution: Print Version is also available. You may find it helpful to have a copy of his solution handy when you work on the Share 2 questions.

 

Share 2

 

After examining Joel’s proof, answer the following questions with a partner or group:

1. Is Joel’s proof valid? Why or why not? If the proof is incorrect, identify the error.
   
   
2. How could a counterexample be used in this situation? Provide an example.

Explore

 

Earlier in this module, you learned that conjectures based on inductive reasoning may be valid for specific examples but cannot be proven. It is impossible to look at all possible cases. Conjectures can be weakened by insufficient evidence or by ignoring counterexamples (e.g., only testing those cases that you know support the conjecture). Conjectures can also be invalidated by a single counterexample. In a similar way, proofs can be invalidated if there are errors in the proof or if invalid assumptions are used when developing the proof. In Lesson 3 you learned how deductive reasoning can be used to prove conjectures. However, not all proofs are valid. Joel’s Solution is an example of an invalid proof.

 

Joel’s proof contained an error. In his solution, Joel did not divide both terms in the numerator by the term in the denominator.

 

 

Dividing both terms in the numerator by the denominator results in n + 1.5.

 

 

Since 6 is not evenly divisible by 4, this counterexample disproves the conjecture that the sum of an even number of consecutive integers is evenly divisible by the number of integers.

 

Sometimes the identification of an error in a proof can be used to correct mistakes in a proof.

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Self-Check 1

 

Self-Check 2

 

Complete “Practising” question 3 on page 42 of the textbook. Answer

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Proofs that include an argument that is based on a false statement are invalid. Even if a valid argument is used, if one or more of the starting premises are not true, the conclusion cannot be true.

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Example 3 is an example of circular reasoning. Circular reasoning uses the conclusion that is to be proved as part of the argument. In other words, you assume that what you are supposed to be proving is true and use that statement as part of your proof. This is like arguing in a circle and is not a valid form of reasoning. The following statements and arguments are examples of circular reasoning.

 

 

1 Adapted from: CANAVAN-MCGRATH ET AL. Principles of Mathematics 11, © 2012 Nelson Education Limited. p. 38. Reproduced by permission.

2 Adapted from: CANAVAN-MCGRATH ET AL. Principles of Mathematics 11, © 2012 Nelson Education Limited. p. 39. Reproduced by permission.

 

Errors in logic can also lead to invalid conclusions. Take a look at the following cartoon.

 

 

It is true that penguins are black and white. It is true that some old television shows are black and white. To deduce from those two statements that penguins are old TV shows is not logical. There is no connection between the two statements. When there is an error in logic (an invalid argument), no reliable conclusion can be made regardless of whether the premises are true.

 

Share 3

 

Consider the following arguments. A Venn diagram can be used to examine each argument.

 

 

With a partner, discuss whether each argument is valid and justify your reasoning. How do the diagrams help you to answer this question? Where does Kai’s name belong in the Venn diagram? Where does Rose’s name belong in the Venn diagram?

 

Self-Check 3

1. Decide whether this argument is valid or invalid. Justify your response:
   
   
   1. Val lives in Sylvan Lake. All people who live in Sylvan Lake like to swim. Therefore, Val likes to swim. Answer
    
   2. A pound of Julia’s Special Blend coffee costs more than $22.75 per pound. John didn’t buy Julia’s Special Blend when he purchased coffee. Therefore, John didn’t spend more than $22.75 per pound. Answer

Connect

 

Lesson Assignment

Lesson 4 Summary

 

In this lesson you discovered that not all proofs are valid. A proof can be invalidated by a single error in reasoning. That error may be the result of division by zero somewhere in the proof. The use of circular reasoning or another type of error in logic also results in an invalid proof. For instance, when reasoning relies on stereotypes, the conclusions are not valid. Stereotypes are generalizations and there are always counterexamples to stereotypes.

 

An error in a proof is similar to a counterexample in that it only takes one error to invalidate the conclusion. For example, a proof that is based on a false premise is invalid, even if the argument used is valid.

 

In Lesson 5 you will apply what you have learned about reasoning to solve problems.