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1. Foundations of Geometry
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United States of AmericaFL
Grade 912

Honors: 1.07 Indirect proof

Lesson
Practice
Lesson

Concept summary

We have seen how we can prove statements using direct reasoning in the form of proofs. We can also disprove statements that are invalid by providing a counterexample.

Counterexample

An example that shows a statement is false

Counterexamples are helpful because proving that a statement is true requires a formal proof while proving that it is false only requires one example where the given statement does not hold true.

We can use the following steps to identify a counterexample:

  1. Identify the condition and conclusion of the statement
  2. Generate options that satisfy the condition
  3. Of those options, any for which the conclusion is false will be a counterexample

We can prove theorems and other deductions indirectly. When we do this, it is referred to as an indirect proof. For mathematical proofs, there are two types of indirect proofs:

Proof by contradiction

A type of proof which starts by assuming that the given information and the negation of the proof goal are true. If this assumption leads to a contradiction, then the original statement is proved true.

A contradiction occurs when two different pieces of information that are assumed to be true, cannot be true at the same time.

Proof by contradiction in steps with symbols:

  1. Assume p and \sim q are true.
  2. Show that assuming \sim q is true leads to a contradiction.
  3. Conclude that q is true.

Worked examples

Example 1

Provide a counterexample to show that the statement "The square of any integer is an even number" is invalid.

Approach

We should first identify the condition and conclusion of the statement, then we can generate a list of options that satisfy that condition, and finally choose one that does not satisfy the conclusion.

The condition is that the number must be the square of an integer. The conclusion is that the result will be an even number.

Solution

Some possible options are 2^2=4, 3^2=9, or 4^2=16. All are squares of integers, but 3^2 results in 9 which is an odd number.

So, 3 is one of many possible counterexamples to the given statement.

Example 2

Suppose that we want to prove the statement "if two integers are odd, then their product will also be odd" is true using proof by contradiction.

a

State the assumptions that we should make for this proof.

Approach

For a proof by contradiction, we want to assume that the given information and the negative of the proof goal are true. In this case, the given information is that "two integers are odd" and the proof goal is that "the product is also odd".

Solution

To prove by contradiction, we assume that:

  • The two integers are odd.
  • The product of the two integers is not odd.

Reflection

We could also word the second assumption as "the product of the two integers is even."

b

Write a proof by contradiction for the statement.

Approach

We want to show that our assumptions in the previous part will lead to a contradiction, thus proving the original statement.

Solution

Given that two integers are odd and assume that their product is not odd. Since the two integers are odd, neither will have a factor of 2. Therefore, the product of the two integers will also not have a factor of 2. However, this contradicts the fact that the product is assumed to be not odd, because not being odd means it is even and has a factor of 2.

Therefore, using proof by contradiction, we have proved that if two integers are odd, then their product will also be odd.

Outcomes

MA.912.GR.1.1

Prove relationships and theorems about lines and angles. Solve mathematical and real-world problems involving postulates, relationships and theorems of lines and angles.

MA.912.LT.4.3

Identify and accurately interpret "if... then", "if and only if", "all" and "not" statements. Find the converse, inverse and contrapositive of a statement.

MA.912.LT.4.8

Construct proofs, including proofs by contradiction.

MA.912.LT.4.10

Judge the validity of arguments and give counterexamples to disprove statements.

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