Question

Prove that the set of rational number is enumerable

Answer 1

Answer:

An easy proof that rational numbers are countable. A set is countable if you can count its elements. Of course if the set is finite, you can easily count its elements. If the set is infinite, being countable means that you are able to put the elements of the set in order just like natural numbers are in order.

Answer 2

Answer:To prove that the set of rational numbers is enumerable

Step-by-step explanation:

To prove that the set of rational numbers is enumerable, we must show that the set can be put into one-to-one correspondence with the set of natural numbers (also called the counting numbers). This means that we can list the rational numbers in such a way that each rational number corresponds to a unique natural number, and vice versa.

One way to do this is to list the rational numbers in increasing order of magnitude. We can start by listing all of the fractions with a numerator of 1 and a denominator of 2, 3, 4, and so on. These are all of the fractions of the form $1/n$, where n is a natural number.

Next, we can list all of the fractions with a numerator of 2 and a denominator of 3, 4, 5, and so on. These are all of the fractions of the form $2/n$, where n is a natural number greater than 1.

We can continue this process for all of the integers, listing the fractions of the form$3/n$, $4/n$ and so on. This will give us a complete list of all of the fractions with a numerator of 1, 2, 3, and so on.

Finally, we can also include the negative fractions in our list by listing the negative versions of each of the fractions we have already listed. For example, we can include the fraction $-1/2$ after the fraction $1/2$, and the fraction $-2/3$ after the fraction $2/3$, and so on.

This process will give us a complete list of all of the rational numbers, and since we have listed them in one-to-one correspondence with the set of natural numbers, we can conclude that the set of rational numbers is enumerable

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