Proof by Exhaustion is a mathematical proof strategy that involves checking all possible cases to prove a statement true. This method is particularly useful when the number of cases is manageable, allowing the mathematician to exhaustively verify each one. Below are three diverse examples that illustrate how this technique can be applied in various contexts.
In number theory, proving whether a set of numbers has certain properties can often be simplified by checking all possible cases. Here, we will prove that the sum of two even numbers is always even.
To do this, we can express even numbers as 2n, where n is an integer. We will check all possible cases for small integers.
When we add two even numbers:
Now, consider all combinations:
After checking all combinations, we see that every sum results in an even number. Therefore, we conclude that the sum of any two even numbers is even.
This proof can be generalized for any integers a and b, reinforcing the conclusion that the sum of two even integers is even. The case-by-case examination demonstrates the effectiveness of Proof by Exhaustion.
In geometry, we might want to prove that all triangles with integer side lengths that sum to 6 can only be isosceles or scalene. We can check all possible combinations of side lengths that meet the criteria of a triangle.
The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side. We will check all combinations of integers that sum to 6:
After checking all combinations of integer values that sum to 6, we find that valid triangles are (1, 2, 3), (2, 2, 2), and (2, 3, 1).
This example shows how Proof by Exhaustion can be effectively used in geometry and reinforces the importance of the triangle inequality. The method also illustrates the limitations of side combinations, leading us to conclude types of triangles possible within the given constraints.
In number theory, modular arithmetic often requires validating properties across a range of integers. Here, we will prove that for integers n from 1 to 5, n^2 modulo 5 results in certain values.
To verify this, we can compute n^2 mod 5 for each integer n in the specified range:
The results are:
This proof illustrates that the possible results of n^2 mod 5 for integers 1 through 5 are 0, 1, and 4. This method showcases how Proof by Exhaustion is effective in modular arithmetic to verify all potential outcomes systematically. The results can be used for further mathematical explorations or applications in cryptography and computer science.