• Devil Moo
In summary, Proof by contradiction starts by supposing a statement, and then shows the contradiction.
Devil Moo
Proof by contradiction starts by supposing a statement, and then shows the contradiction.

1. Homework Statement

Now, there is a statement ##A##.
Suppose ##A## is false.
So ##A## is true.

My question:
There are two statements ##A## and ##B##.
Suppose ##A## is true.
Further suppose ##B## is false.
Can I conclude that "if ##A## is true, then ##B## is false."

Moreover, how do I distinguish between "Proof by Contradiction" and "Proof by Exhaustion"?

Devil Moo said:
Can I conclude that "if A is true, then B is false."
No. But you can conclude that if A is true then B is true.
Devil Moo said:
Moreover, how do I distinguish between "Proof by Contradiction" and "Proof by Exhaustion"?
Proof by exhaustion does not necessarily involve any contradictions. Here's the definition from wiki:
Proof by exhaustion, also known as proof by cases, proof by case analysis, perfect induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases and each type of case is checked to see if the proposition in question holds.

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Devil Moo said:
Moreover, how do I distinguish between "Proof by Contradiction" and "Proof by Exhaustion"?
Here's an example of a proof by exhaustion.
Assume that n is a positive integer. Then n(n + 1)(n + 2) is divisible by 3.

Case 1: n = 3k, for some integer k
Then n(n + 1)(n + 2) = 3k(3k + 1)(3k + 2), which clearly has a factor of 3, and so is divisible by 3

Case 2: n = 3k + 1, for some integer k
Then n(n + 1)(n + 2) = (3k + 1)(3k + 2)(3k + 3) = (3k + 1)(3k + 2)3(k + 1), which has a factor of 3, so is divisible by 3.

Case 3: n = 3k + 2, for some integer 8
Then n(n + 1)(n + 2) = (3k + 2)(3k + 3)(3k + 4) = (3k + 2)3(k + 1)(3k + 4), also has a factor of 3, so is divisible by 3.

There are no other possible cases. Any integer n falls into one of three equivalence classes: ##n \equiv 0 (\mod 3)##, or ##n \equiv 1 (\mod 3)##, or ##n \equiv 2 (\mod 3)##. IOW, when an integer n is divided by 3, the remainder will be 0, 1, or 2. The three cases above are based on this fact.

Reopened...

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## What is "Proof by Contradiction"?

"Proof by Contradiction" is a type of mathematical proof that involves assuming the opposite of what you want to prove and showing that it leads to a contradiction. This contradiction then proves that the original statement must be true.

## How does "Proof by Contradiction" work?

The first step in "Proof by Contradiction" is to assume the opposite of the statement you want to prove. Then, you use logical reasoning and mathematical principles to show that this assumption leads to a contradiction. This contradiction then proves that the original statement must be true.

## Why is "Proof by Contradiction" useful in mathematics?

"Proof by Contradiction" is useful in mathematics because it allows us to prove statements that may be difficult to prove directly. It also helps us to better understand the logical connections between different mathematical concepts and principles.

## What are the limitations of "Proof by Contradiction"?

One limitation of "Proof by Contradiction" is that it may not always be applicable or effective in proving a statement. It also requires a high level of logical reasoning and can be time-consuming.

## Can "Proof by Contradiction" be used in other fields besides mathematics?

Yes, "Proof by Contradiction" can be used in other fields such as philosophy and logic. It involves using logical reasoning to show that a statement or argument is false, which can be applied in various disciplines.

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