Constructing proofs of denumerable sets

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However, if the elements of the set can be represented by a table of infinite elements, you can enumerate the elements row by row, even if the number of elements in each row is infinite. This is sufficient to prove that the set is denumerable. In summary, when proving that a set is denumerable, it is necessary to show a one-to-one correspondence with the set of natural numbers, but the method of proof may vary depending on the type of set being studied.
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buffordboy23
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Hi, I am doing some self-studying on the topic of functional analysis, specifically set theory at the moment.

Suppose that we want to show that some set is denumerable. Is it required that we directly show the one-to-one correspondence between the elements of the set and the set of natural numbers? This seems to be the method of proof employed in the text for the given theorems, but the results of the theorems themselves seem to offer hand-waving. For example, the set that consists of the sum of a denumerable number of denumerable sets is itself denumerable.

Now suppose that the elements of some set that we are trying to prove is denumerable can be represented by a table of infinite elements. Must we use the "diagonal method" to prove that we can enumerate the elements of this set? Or is it sufficient to enumerate row by row, although the number of elements in each row of the table is infinite (we would never make it to the next row)?
 
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HI

when constructing proofs for a denumerable number of denumerable sets you will need to use diagonal counting.
 

Related to Constructing proofs of denumerable sets

What is a denumerable set?

A denumerable set is a set that can be put in one-to-one correspondence with the set of natural numbers (1, 2, 3, ...). This means that each element in the set can be assigned a unique natural number.

What is the importance of constructing proofs of denumerable sets?

Constructing proofs of denumerable sets is important in mathematics because it allows us to understand the properties and characteristics of infinite sets. It also helps us to establish the existence and uniqueness of certain objects within these sets.

What are some common methods used to construct proofs of denumerable sets?

Some common methods used to construct proofs of denumerable sets include using bijections, creating sequences, and using mathematical induction.

How can we prove that a set is denumerable?

To prove that a set is denumerable, we need to show that it can be put in one-to-one correspondence with the set of natural numbers. This can be done by explicitly constructing a bijection between the two sets or by showing that the set can be enumerated in a systematic way.

What are some examples of denumerable sets?

Some examples of denumerable sets include the set of all positive integers, the set of even numbers, and the set of rational numbers. These sets can all be put in one-to-one correspondence with the set of natural numbers.

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