Set notation: {x∈ℕ∣x+3∈ℕ} is evidently equivalent to {0,1,2,3...}...?

[Aion]

Set notation: {x∈ℕ∣x+3∈ℕ} is evidently equivalent to {0,1,2,3...}...?

{\(\displaystyle x \in \mathbb{N}\mid x+3 \in\mathbb{N} \)} This is the set of all natural numbers which are 3 less than a natural number.

This is apprently equivalent to {\(\displaystyle 0, 1, 2, 3 ... \)}.


Can anyone please explain why we allow \(\displaystyle x\) to be equal to \(\displaystyle -3\)? Clearly \(\displaystyle x \in\mathbb{N}\) and if \(\displaystyle x = -3\) then \(\displaystyle x\notin\mathbb{N}\).

I would've read this statement as "The set of all \(\displaystyle x\) in the natural numbers such that \(\displaystyle x + 3\) is an element of the natural numbers." And written it as {\(\displaystyle 3, 4, 5, 6 ... \)}.

 

[Dr.Peterson]

{\(\displaystyle x \in \mathbb{N}\mid x+3 \in\mathbb{N} \)} This is the set of all natural numbers which are 3 less than a natural number.

This is apprently equivalent to {\(\displaystyle 0, 1, 2, 3 ... \)}.

Can anyone please explain why we allow \(\displaystyle x\) to be equal to \(\displaystyle -3\)? Clearly \(\displaystyle x \in\mathbb{N}\) and if \(\displaystyle x = -3\) then \(\displaystyle x\notin\mathbb{N}\).

I would've read this statement as "The set of all \(\displaystyle x\) in the natural numbers such that \(\displaystyle x + 3\) is an element of the natural numbers." And written it as {\(\displaystyle 3, 4, 5, 6 ... \)}.

Apparently you are working with the definition of N as {0, 1, 2, 3, ...}; often it is defined as {1, 2, 3, ...}.

You can't include -3, because that is not in N, and the set specifies that x is in N. But the answer you say you were given does not include -3, so I don't know why you are asking.

Every number in N is in the specified set, because 3 more than any natural number is still a natural number (3 more than 0 is 3, and so on).

So the answer is just N.

In excluding 0, 1, and 2 from your answer, you are forgetting that 3 more than each of those is in N. It looks as if you are thinking of the set of natural numbers that are 3 more than a natural number -- that is, x - 3 is in N. Somehow, you are reading things backward.

 

[Aion]

Apparently you are working with the definition of N as {0, 1, 2, 3, ...}; often it is defined as {1, 2, 3, ...}.

You can't include -3, because that is not in N, and the set specifies that x is in N. But the answer you say you were given does not include -3, so I don't know why you are asking.

Every number in N is in the specified set, because 3 more than any natural number is still a natural number (3 more than 0 is 3, and so on).

So the answer is just N.

In excluding 0, 1, and 2 from your answer, you are forgetting that 3 more than each of those is in N. It looks as if you are thinking of the set of natural numbers that are 3 more than a natural number -- that is, x - 3 is in N. Somehow, you are reading things backward.

Yeah, I've been reading it backwards all along.

{\(\displaystyle x \in \mathbb{N}\mid x+3 \in\mathbb{N} \)}

Let \(\displaystyle x = 0\), then \(\displaystyle 0+3 = 3\in\mathbb{N}\implies x=0\) satisfy the condition.
Let \(\displaystyle x = 1\), then \(\displaystyle 1+3 = 4\in\mathbb{N}\implies x=1\) satisfy the condition.
Let \(\displaystyle x = 2\), then \(\displaystyle 2+3 = 5\in\mathbb{N}\implies x=2\) satisfy the condition.
etc...

Then the set of all \(\displaystyle x\) which satisfy the condition is {\(\displaystyle 0, 1 ,2, 3 ... \)} = \(\displaystyle \mathbb{N}\)

 

[Steven G]

Yeah, I've been reading it backwards all along.

{\(\displaystyle x \in \mathbb{N}\mid x+3 \in\mathbb{N} \)}

Let \(\displaystyle x = 0\), then \(\displaystyle 0+3 = 3\in\mathbb{N}\implies x=0\) satisfy the condition.
Let \(\displaystyle x = 1\), then \(\displaystyle 1+3 = 4\in\mathbb{N}\implies x=1\) satisfy the condition.
Let \(\displaystyle x = 2\), then \(\displaystyle 2+3 = 5\in\mathbb{N}\implies x=2\) satisfy the condition.
etc...

Then the set of all \(\displaystyle x\) which satisfy the condition is {\(\displaystyle 0, 1 ,2, 3 ... \)} = \(\displaystyle \mathbb{N}\)

. As Dr P stated 0 is NOT in N. N is also known as the set of Counting Numbers and when we count we start with 1. N= {1,,3,...}. On the other hand W, the set of Whole Numbers, start with 0. W = {0, 1, 2, 3,...}.

 

[Dr.Peterson]

. As Dr P stated 0 is NOT in N. N is also known as the set of Counting Numbers and when we count we start with 1. N= {1,,3,...}. On the other hand W, the set of Whole Numbers, start with 0. W = {0, 1, 2, 3,...}.

Actually, I said nothing of the sort. I said that sources vary: see http://mathworld.wolfram.com/NaturalNumber.html,

The term "natural number" refers either to a member of the set of positive integers 1, 2, 3, ... or to the set of nonnegative integers 0, 1, 2, 3, ... . Regrettably, there seems to be no general agreement about whether to include 0 in the set of natural numbers.

The set of natural numbers (whichever definition is adopted) is denoted N.
​

Presumably the OP was taught the alternative definition, so we work with that.

 

[Steven G]

Actually, I said nothing of the sort. I said that sources vary: see http://mathworld.wolfram.com/NaturalNumber.html,

The term "natural number" refers either to a member of the set of positive integers 1, 2, 3, ... or to the set of nonnegative integers 0, 1, 2, 3, ... . Regrettably, there seems to be no general agreement about whether to include 0 in the set of natural numbers.

The set of natural numbers (whichever definition is adopted) is denoted N.
​

Presumably the OP was taught the alternative definition, so we work with that.

Yes, it is correct that you did not say what I claimed you said. To be honest I thought that you were just being polite about the way you responded. I hope that my post did not offend you.

 

[Dr.Peterson]

Yes, it is correct that you did not say what I claimed you said. To be honest I thought that you were just being polite about the way you responded. I hope that my post did not offend you.

I am not offended. But, yes, I was being polite (by allowing for other perspectives).

I've had a lot of experience with people around the world, and have found that a lot of things I think I know are not universally agreed to. This was a chance to state explicitly what I had only implied.