Is set of points (p,q) countable?

In summary, the conversation discusses the countability of the set of points on a plane with rational coordinates. One person has already proven the countability of the set of rational numbers through a table drawing, but is unsure how to combine this with the proof for the set of points. The other person suggests using a mathematical proof instead of a table drawing, stating that any subset of the rationals is either finite or countable since the countability of the rationals has already been proven."
  • #1
seyma
8
0
I want to show that the set of points (p,q) on the plane with rational coordinates p and q is countable. I proved set of rational numbers is countable by drawing table and I find (http://web01.shu.edu/projects/reals/infinity/proofs/combctbl.html ) Combining Countable Sets. However, I cannot put these together in table.
Could it be proven by matematically instead of table drawing?
 
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  • #2
If you know how to prove that there are only countably many rationals, the proof is exactly the same.
 
  • #3
For that matter, since you have proven that there are only countably many rationals, any subset of the rationals is either finite or countable.
 

1. What does it mean for a set of points to be countable?

A set of points is considered countable if there exists a one-to-one correspondence between the points in the set and the natural numbers (1, 2, 3, ...). This means that every point in the set can be assigned a unique number, and there are no points left out.

2. Is it possible for an infinite set of points to be countable?

Yes, it is possible for an infinite set of points to be countable. As long as there exists a one-to-one correspondence between the points in the set and the natural numbers, the set can be considered countable.

3. How do you prove that a set of points is countable?

To prove that a set of points is countable, you can use a technique called "counting by diagonals". This involves creating a table with the points in the set listed in rows and columns, and then drawing diagonals to show the one-to-one correspondence with the natural numbers. If every point in the set can be paired with a natural number in this way, then the set is countable.

4. Are all sets of points countable?

No, not all sets of points are countable. There are sets of points that are uncountable, meaning that there is no one-to-one correspondence between the points and the natural numbers. A common example of an uncountable set is the set of real numbers.

5. What applications does the concept of countable sets have in mathematics?

The concept of countable sets is important in many areas of mathematics, including analysis, topology, and number theory. It is also used in computer science and cryptography. For example, the countability of a set can determine the solvability of certain mathematical problems, and the properties of countable sets can be used to prove theorems and develop new mathematical concepts.

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