To prove a theorem indirectly, we begin by assuming that its...

Name: Problem 2, Chapter 1: Geometry
Uploaded: 2020-08-13T15:44:56-08:00
Duration: 38 s
Channel: Ian Shi
Description: To prove a theorem indirectly, we begin by assuming that its conclusion is false. List the assumption with which an indirect proof of each of the following statements would begin. Example: If a tailor wants to make a coat last, he makes the pants first. Answer: Suppose that the tailor does not make the pants first. If a teacher is cross-eyed, he has no control over his pupils.

To prove a theorem indirectly, we begin by assuming that its conclusion is false. List the assumption with which an indirect proof of each of the following statements would begin. Example: If a tailor wants to make a coat last, he makes the pants first. Answer: Suppose that the tailor does not make the pants first. If a teacher is cross-eyed, he has no control over his pupils.

To prove a theorem indirectly, we begin by assuming that its conclusion is false. List the assumption with which an indirect proof of each of the following statements would begin.
Example: If a tailor wants to make a coat last, he makes the pants first.
Answer: Suppose that the tailor does not make the pants first.
If a teacher is cross-eyed, he has no control over his pupils.

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Key Concepts

Conditional Statements

Conditional statements are propositions of the form 'if P, then Q'. Understanding these statements is key in indirect proofs because to negate a conditional statement, one typically assumes that P is true and Q is false. This approach is fundamental when applying indirect proof strategies, as it directly addresses the logical structure of many theorems and propositions.

Indirect Proof

Indirect proof (also known as proof by contradiction) is a method of establishing the truth of a statement by assuming the opposite (i.e., assuming that the statement is false) and then showing that this assumption leads to an inconsistency or contradiction. This contradiction implies that the initial assumption was incorrect, thereby confirming the truth of the original statement.

Negation of the Conclusion

In an indirect proof, the process begins by assuming the negation of the conclusion. This is a critical step because it sets up the scenario from which a contradiction is derived. For a conditional statement (an 'if-then' statement), this involves assuming that the condition holds while the conclusion does not, which is essential in arriving at the contradiction needed to validate the original statement.

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