Cardinality of infinite subset of infinite set

In summary: N} injects into \mathbb{Q} (and in fact one can show that this map is surjective).In summary, cardinality refers to the size or amount of elements in a set. If one set is a subset of another set, then the cardinality of the first set is less than or equal to the cardinality of the second set. This is true for all sets, including the subsets of the sets of integers and rational numbers. While the cardinality of the integers may seem smaller than the cardinality of the rationals, there exists a bijection or one-to-one correspondence between the two sets, showing that they have the same cardinality. Therefore, the
  • #1
Bipolarity
776
2
Am a bit confused about the meaning of cardinality. If ## A \subseteq B ##, then is it necessarily the case that ## |A| \leq |B| ##?

I am thinking that since ## A \subseteq B ##, an injection from A to B exists, hence its cardinality cannot be greater than that of B?

But this cannot be correct, since ##\mathbb{Z}## and ##\mathbb{Q}## have the same cardinality?

Where am I wrong?

Thanks!

BiP
 
Physics news on Phys.org
  • #2
You are right. If ##A\subseteq B##, then ##|A|\leq |B|##.

In particular, ##|\mathbb{Z}|\leq |\mathbb{Q}|## is true. Don't confuse this with ##|\mathbb{Z}|< |\mathbb{Q}|##, which is false.
 
  • #3
R136a1 said:
You are right. If ##A\subseteq B##, then ##|A|\leq |B|##.

In particular, ##|\mathbb{Z}|\leq |\mathbb{Q}|## is true. Don't confuse this with ##|\mathbb{Z}|< |\mathbb{Q}|##, which is false.

I see. So which of the following is true?
##|\mathbb{Z}|< |\mathbb{Q}|##
##|\mathbb{Z}|= |\mathbb{Q}|##

Thanks!

BiP
 
  • #5


Cardinality refers to the size or number of elements in a set. In this context, the cardinality of a set is considered to be infinite if it has an infinite number of elements. The cardinality of a set can be compared to the cardinality of another set by looking at the existence of a one-to-one correspondence, or bijection, between the elements of the two sets.

In the case of an infinite subset of an infinite set, the cardinality of the subset cannot be greater than the cardinality of the larger set. This is because, as you mentioned, there exists an injection from the subset to the larger set, meaning there is a way to map each element in the subset to a unique element in the larger set. This shows that the cardinality of the subset is less than or equal to the cardinality of the larger set.

However, as you also pointed out, this does not necessarily mean that the cardinality of the subset is strictly smaller than the cardinality of the larger set. In the case of the sets ##\mathbb{Z}## and ##\mathbb{Q}##, they have the same cardinality because there exists a bijection between the two sets. This shows that while the subset cannot have a greater cardinality than the larger set, it can have the same cardinality.

In summary, the cardinality of an infinite subset of an infinite set is always less than or equal to the cardinality of the larger set, but it can be equal in some cases. This is due to the existence of bijections between sets with the same cardinality, even if one set is a subset of the other.
 

What is the definition of "cardinality" in mathematics?

In mathematics, the cardinality of a set is the number of elements in the set. It is a measure of the size or magnitude of a set.

Can an infinite subset of an infinite set have the same cardinality as the original set?

Yes, it is possible for an infinite subset of an infinite set to have the same cardinality as the original set. This is known as a "countably infinite" set, where the elements of the subset can be put into a one-to-one correspondence with the elements of the original set.

How do you prove that the cardinality of an infinite subset is the same as the original set?

To prove that two sets have the same cardinality, you need to show that there is a one-to-one correspondence between the elements of the two sets. This means that each element in one set corresponds to exactly one element in the other set, and vice versa.

Are there different levels of infinity when it comes to infinite sets?

Yes, there are different levels of infinity when it comes to infinite sets. For example, the set of natural numbers (1, 2, 3, ...) is countably infinite, while the set of real numbers is uncountably infinite.

What is the significance of understanding the cardinality of infinite subsets of infinite sets?

Understanding the cardinality of infinite subsets is important in many areas of mathematics, including set theory, calculus, and analysis. It helps us to better understand the properties and behaviors of infinite sets, and allows us to make mathematical arguments and proofs about them.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
19
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
16
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
19
Views
3K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
20
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
12
Views
8K
Back
Top