# Elements in sets

Math100
Homework Statement:
Write each of the following sets by listing their elements between braces.
Relevant Equations:
None.
Can anyone please check/confirm my answers if they are correct or not? I boxed around my answers just to be clear and understanding. Thank you.

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Homework Statement:: Write each of the following sets by listing their elements between braces.
Relevant Equations:: None.

Can anyone please check/confirm my answers if they are correct or not? I boxed around my answers just to be clear and understanding. Thank you.
They look fine.

Delta2 and Math100
Math100

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Yes, although any thoughts on whether the second question is valid?

Math100
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It looks ok. For (2), some people consider 0 to be a natural number.

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It depends whether the notation, ##\{x \in \mathbb N: \dots \}## implies that ##\mathbb N## is the universal set under consideration. In that case, ##-2 \notin \mathbb N## and the comparison ##-2 < x## is not valid.

Or, if we consider that in all cases ##\mathbb N \subset \mathbb Z## and that it's valid to talk about ##-2## even when nominally restricting our attention to ##\mathbb N##, then it's fine.

I'm not saying one way or the other, but the question just didn't look right to me.

Math100
Thank you guys for the help! I really appreciate it!

Delta2
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Thank you guys for the help! I really appreciate it!
What do you think? Is the condition ##-2 < x## valid for ##x \in \mathbb N##?

Math100
What do you think? Is the condition ##-2 < x## valid for ##x \in \mathbb N##?
Yes.

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It depends whether the notation, ##\{x \in \mathbb N: \dots \}## implies that ##\mathbb N## is the universal set under consideration. In that case, ##-2 \notin \mathbb N## and the comparison ##-2 < x## is not valid.
By that reasoning, it would not be possible to say whether "1<x" is valid, since we could take the 1 as an element of that ##\mathbb Z## or ##\mathbb R##.
Similarly, I could not write x-1 since that is shorthand for x+(-1).

Seems more reasonable to apply the programming language concept of type coercion. An element of ##\mathbb N## can be 'elevated' to ##\mathbb Z## etc. as necessary to make the operation valid.

Whether the result can be demoted to conform to the target variable type is another matter.

Delta2
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By that reasoning, it would not be possible to say whether "1<x" is valid, since we could take the 1 as an element of that ##\mathbb Z## or ##\mathbb R##.
Similarly, I could not write x-1 since that is shorthand for x+(-1).

Seems more reasonable to apply the programming language concept of type coercion. An element of ##\mathbb N## can be 'elevated' to ##\mathbb Z## etc. as necessary to make the operation valid.

Whether the result can be demoted to conform to the target variable type is another matter.
Usually ##\mathbb R## is implied at the universal set and ##\mathbb Z \subset \mathbb R##.

The point about the question is that it explicitly defines the universal set as ##\mathbb N## and then talks about ##-2##, which is not defined within ##\mathbb N##.

Anyway, my main point is that it's definitely worth thinking about if you want to study pure maths,

Delta2