Question

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 proof is indirect, then it leads to a contradiction.

Name: Problem 3, Chapter 1: Geometry
Uploaded: 2020-08-13T15:46:01-08:00
Duration: 39 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 proof is indirect, then it leads to a contradiction.

   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 proof is indirect, then it leads to a contradiction.

Geometry

Harold R. Jacobs 2nd Edition

Chapter 1, Problem 3

Instant Answer

Solved by Expert Ian Shi

08/13/2020

Step 1

In this case, we are trying to prove that "If a proof is indirect, then it leads to a contradiction." So, we assume the opposite: "Suppose a proof is indirect, but it does not lead to a contradiction." Show more…

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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 proof is indirect, then it leads to a contradiction.

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Key Concepts

Contradiction

In the context of an indirect proof, a contradiction refers to a pair of logically incompatible statements that arise from the assumed negation of the conclusion. The presence of such a contradiction signifies that the initial assumption must be false, thus confirming the original theorem.

Assumption of Negation

The initial step in an indirect proof involves assuming the exact negation of the conclusion. This assumption is used as a starting point to explore the logical consequences, with the intent of revealing inconsistencies or contradictions.

Indirect Proof

A method of proving a theorem by assuming that the conclusion is false and then demonstrating that this assumption leads logically to a contradiction, thereby establishing the truth of the original statement.

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