Proof by exhaustion is a mathematical method used to establish the truth of statements by dividing them into a finite number of cases and verifying each one. This technique is particularly effective for propositions that can be categorized into well-defined scenarios, such as properties dependent on number parity or congruence. It involves identifying all possible cases and then proving the conjecture for each, ensuring a robust validation of the statement.
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This method involves checking the truth of a statement for each possible case
Differences from Proof by Exhaustion
Proof by exhaustion relies on the comprehensive verification of each specific instance, while general logical arguments use more abstract reasoning
Proof by exhaustion is particularly effective for properties that depend on the parity of numbers or the congruence of integers modulo a number
The first step involves carefully and exhaustively enumerating every possible scenario that must be considered
The second step is to analyze and prove the conjecture for each case, ensuring a meticulous examination of all possibilities
If the conjecture holds for all cases, the proof is complete, but if any case does not satisfy the conjecture, it is disproven
Proof by exhaustion is most suitable when the conjecture can be broken down into a finite and manageable number of cases
This method is not practical for conjectures with an infinite or very large number of cases
Proof by exhaustion is particularly useful in discrete mathematics, where the elements under consideration are countable and the cases are well-defined
The conjecture that for any integer \(n\) between 2 and 7, the expression \(n^2 + 2\) is not a multiple of 4 can be proven by evaluating the expression for each integer value of \(n\) within the range
The statement that for any prime number \(p\) between 3 and 25, the product \((p - 1)(p + 1)\) is divisible by 12 can be confirmed by examining each prime number within the specified range
The divisibility of \(n^7-n\) by 7 for all positive integers \(n\) can be confirmed by considering cases where \(n\) is a multiple of 7 or when \(n\) has a remainder of 1 through 6 upon division by 7
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The method of ______ by ______ is not applicable for infinite scenarios since checking each case would be unfeasible.
proof
exhaustion
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Proof by exhaustion: Step 1 detail
Identify all distinct cases for the conjecture, ensuring no scenario is overlooked.
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Proof by exhaustion: Step 2 detail
Verify conjecture for each identified case, proving its truth individually.
