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Questions and Answers
What is the definition of an indirect proof?
What is the definition of an indirect proof?
The method of proving by assuming the conclusion is false, leading to a contradiction.
The indirect proof method is sometimes called the proof of contradiction.
The indirect proof method is sometimes called the proof of contradiction.
True (A)
What is the first step in proving a statement using indirect proof?
What is the first step in proving a statement using indirect proof?
- Prove the statement directly
- Assume the original statement is true
- Use an example to demonstrate
- Assume the opposite of the conclusion (correct)
What follows after assuming the opposite of the conclusion in an indirect proof?
What follows after assuming the opposite of the conclusion in an indirect proof?
What conclusion is reached if a contradiction is found in an indirect proof?
What conclusion is reached if a contradiction is found in an indirect proof?
What is the proposition demonstrated in Example 2.1?
What is the proposition demonstrated in Example 2.1?
What assumption is made in Example 2.2 for the indirect proof?
What assumption is made in Example 2.2 for the indirect proof?
In Example 2.2, the final conclusion is that n is an odd integer.
In Example 2.2, the final conclusion is that n is an odd integer.
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Study Notes
Indirect Proofs
- Indirect proof is a method used to establish the truth of a mathematical statement by assuming the opposite is true.
- This technique is often referred to as proof by contradiction.
- The process involves showing that the assumption leads to a contradiction, which confirms the original claim's validity.
Types of Proofs
- There are two main types of proofs: direct and indirect.
- Direct Proof: Proves the statement directly by a logical chain from premises to conclusion.
- Indirect Proof: Starts by assuming the negation of the conclusion and seeking a contradiction.
Steps in Indirect Proof
- Assume the opposite of the conclusion of the statement you want to prove.
- Proceed under the assumption to derive logical consequences.
- Identify a contradiction arising from this assumption.
- Conclude that the original statement must be true because assuming it was false led to a contradiction.
Propositions and Examples
-
Proposition: "If ( x ) is an odd integer, then ( x ) is an odd integer."
- This is proven directly, validating the claim through the definition of odd integers.
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Proposition: "If the square of an integer ( n ) is even, then ( n ) itself is even."
- Assume ( n ) is odd. Deriving contradictions proves that ( n ) must be even if ( n^2 ) is even.
Key Definitions
- Odd Integer: An integer of the form ( 2p + 1 ), where ( p \in \mathbb{Z} ).
- Even Integer: An integer of the form ( 2p ), where ( p \in \mathbb{Z} ).
General Approach in Examples
- Use variables instead of specific numbers to maintain generality.
- Structure the argument clearly with assumptions, reasoning, and conclusions.
- When proving statements mathematically, denote known premises and conclusions clearly to enhance understanding.
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