a) List the next three numbers after 5 that we would encounter if we were to continue the...

a) List the next three numbers after 5 that we would encounter if we were to continue the listing of the positive rational numbers that we began in Example.

The Set of Rational Numbers Is Countable

Show that the set of rational numbers is countable.

SOLUTION:

First consider the positive rational numbers as we list them in The first row of the arrangement has all positive rational numbers with denominator 1. The second row has all denominators 2, the third row has denominators of 3, and so on. We then trace through the arrangement following the red line skipping over numbers that we have encountered earlier.

We will follow the path through the rational numbers in and list those numbers in a straight line, matching them with the natural numbers as follows:

Notice that in our matching, we have skipped over rational numbers that we have encountered earlier, such as 2/2, 2/4, 3/3, and 4/2.

When we look at the positive rational numbers in this way, we see that there is a first number, then a second, then a third, and so on, and we are listing each number exactly once. Therefore, there is a one-to-one correspondence between N and the positive rational numbers.

To show that the entire set of rational numbers is countable, we would argue as we did to account for zero and the negative rationals; however, we will not go into the details of how to do it. Therefore, the set of rational numbers has the same cardinal number as the natural numbers, which is ℵ0.

The Set of Integers Is Countable

Show that I, the set of integers, can be put in a one-to-one correspondence with the set of natural numbers.

SOLUTION:

The following matching shows the correspondence. (In the correspondence, we are assuming n represents a positive integer.)

Therefore, n(I) = ℵ0, and I is a countable set.

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b) What is the next number that we would skip after 4/2?