# Can b be an integer if it does not divide k for every natural number k?

• pjgrah01
In summary, the task is to prove by contradiction that if b is an integer that does not divide any natural number k, then b must equal 0. The initial attempt at a solution is to assume that "b does not divide k" means "there exists a natural number k that b does not divide". However, this interpretation is ambiguous and the correct approach would be to assume that "b does not divide k" means "there exists a natural number k such that b does not divide any natural number." The contradiction would then be that there exists a non-zero integer b that does not divide any natural number, which leads to the conclusion that b must equal 0.
pjgrah01

## Homework Statement

Prove by contradiction that if b is an integer such that b does not divide k for every natural number k, then b=0.

## The Attempt at a Solution

I know that proof by contradiction begins by assuming the false statement: If b is an integer such that b does not divide k for every kεℕ, then b≠0, which is equivalent to "there exists an integer b such that b does not divide k and b≠0, for every kεℕ. But I'm not sure how to proceed from here.

pjgrah01 said:
I know that proof by contradiction begins by assuming the false statement: If b is an integer such that b does not divide k for every kεℕ, then b≠0
That is not the opposite statement.
"If a pen is green, then it is my pen" is wrong, but "if a pen is green, then it is not my pen" is also wrong (because I own some green pens, but not all).

Also "b does not divide k for every natural number k" is itself ambiguous. It could be read as "b does not divide any natural number" but here, I think, is intended to say "there exist a natural number, k, that b does not divide". The "contradiction" would just be "there exist b, not equal to 0, such that b does not divide any natural number, k." Given that, what can you say about k= 2b?

## 1. What is "Proof by Contradiction"?

Proof by Contradiction is a method of mathematical proof in which we assume the negation of what we want to prove, and then show that this assumption leads to a contradiction. This contradiction implies that the negation of our original statement must be false, therefore proving our original statement to be true.

## 2. How is "Proof by Contradiction" different from other types of proofs?

Proof by Contradiction is different from other types of proofs because it relies on the assumption of the negation of the statement being proved. This allows us to directly reach a contradiction, rather than having to go through a series of logical steps.

## 3. What types of problems can "Proof by Contradiction" be used for?

Proof by Contradiction can be used for a variety of problems in mathematics, including proving theorems, solving equations, and showing the existence or non-existence of certain mathematical structures.

## 4. What are the steps involved in a "Proof by Contradiction"?

The steps involved in a "Proof by Contradiction" are as follows: 1) Assume the negation of the statement to be proved. 2) Use this assumption to reach a contradiction. 3) Conclude that the negation of the original statement must be false, and therefore the original statement is true.

## 5. What are the benefits of using "Proof by Contradiction"?

One of the main benefits of using "Proof by Contradiction" is that it can often lead to shorter and more elegant proofs compared to other methods. It also allows us to prove a statement without having to explicitly state a direct proof, which can be useful in certain situations where a direct proof may be difficult to formulate.

Replies
12
Views
1K
Replies
5
Views
498
Replies
1
Views
787
Replies
3
Views
955
Replies
1
Views
2K
Replies
1
Views
1K
Replies
18
Views
2K
Replies
16
Views
1K
Replies
4
Views
1K
Replies
2
Views
2K