Elements in Sets: Check/Confirm Answers

In summary: 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, since there is a lot of theory and not a lot of practice.If you are dealing with natural numbers, that can't be right, as natural numbers have no additive inverse. Instead ##n - m## is defined to be the natural number ##k## such
  • #1
Math100
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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.
 

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  • #2
Math100 said:
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.
 
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  • #3
So my answers are correct?
 
  • #4
Math100 said:
So my answers are correct?
Yes, although any thoughts on whether the second question is valid?
 
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  • #5
It looks ok. For (2), some people consider 0 to be a natural number.
 
  • #6
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.
 
  • #7
Thank you guys for the help! I really appreciate it!
 
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  • #8
Math100 said:
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##?
 
  • #9
PeroK said:
What do you think? Is the condition ##-2 < x## valid for ##x \in \mathbb N##?
Yes.
 
  • #10
PeroK said:
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.
 
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  • #11
haruspex said:
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,
 
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  • #12
haruspex said:
Similarly, I could not write x-1 since that is shorthand for x+(-1).
If you are dealing with natural numbers, that can't be right, as natural numbers have no additive inverse. Instead ##n - m## is defined to be the natural number ##k## such that ##m + k = n##.

In general, ##n - m## is not well defined for all pairs of natural numbers. That IS important.
 

Related to Elements in Sets: Check/Confirm Answers

1. What are elements in sets?

Elements in sets refer to the individual objects or numbers that are included in a set. A set is a collection of distinct elements that are grouped together based on a specific criteria.

2. How do you check if an element is in a set?

To check if an element is in a set, you can use the "in" operator in Python or the "includes()" method in JavaScript. Both of these methods will return a boolean value indicating whether the element is present in the set or not.

3. Can an element be repeated in a set?

No, by definition, a set does not allow for duplicate elements. If an element is already present in a set, it will not be added again. This is what makes sets different from other data structures, such as lists or arrays.

4. How do you confirm if two sets are equal?

To confirm if two sets are equal, you can use the "==" operator in Python or the "isEqual()" method in JavaScript. These methods will compare the elements in both sets and return a boolean value indicating if they are equal or not.

5. What is the cardinality of a set?

The cardinality of a set refers to the number of elements in that set. It is also known as the "size" of a set. The cardinality of a set can be determined by counting the number of distinct elements in that set.

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