5.8. Indirect Proof in Algebra and Geometry http://www.ck12.org
5.8 Indirect Proof in Algebra and Geometry
Here you’ll learn how to write indirect proofs, or proofs by contradiction, by assuming a hypothesis is false.
What if you know something is true but cannot figure out how to prove it directly? After completing this Concept,
you’ll be able to indirectly prove a statement by way of contradiction.
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Click image to the left for more content.CK-12 Foundation: Chapter5IndirectProofA
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Click image to the left for more content.James Sousa:Introduction toIndirect Proof
Guidance
Until now, we have proved theorems true by direct reasoning, where conclusions are drawn from a series of facts
and previously proven theorems. However, we cannot always use direct reasoning to prove every theorem.
Indirect Proof or Proof by Contradiction:When the conclusion from a hypothesis is assumed false (or opposite
of what it states) and then a contradiction is reached from the given or deduced statements.
In other words, if you are trying to show that something is true, show that if it was not true there would be a
contradiction (something else would not make sense).
The steps to follow when proving indirectly are:
- Assume theoppositeof the conclusion (second half) of the statement.
- Proceed as if this assumption is true to find thecontradiction.
- Once there is a contradiction, the original statement is true.
- DO NOT use specific examples.Use variables so that the contradiction can be generalized.
The easiest way to understand indirect proofs is by example.