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

Aion

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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 {\(\displaystyle 3, 4, 5, 6 ... \)}.
 
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Dr.Peterson

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

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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 {\(\displaystyle 0, 1 ,2, 3 ... \)} = \(\displaystyle \mathbb{N}\)
 

Jomo

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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

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. 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.
 

Jomo

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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

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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.
 
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