# Elements in Sets: Check/Confirm Answers

• Math100
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
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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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.

Delta2 and Math100

Math100 said:
Yes, although any thoughts on whether the second question is valid?

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

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.

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

Delta2
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##?

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

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.

Delta2
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,

Delta2
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.

## 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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