The Number of Points on Two Line Segments

We say that a set is countably infinite if we can pair the elements with set of counting numbers 1, 2, 3, and so on. Believe it or not, the number of positive integers and the number of integers (both negative and positive including 0) have the same number of elements. It is because we can pair them in a one-to-one correspondence such as shown in the below.

As shown on the table, if we continue indefinitely, we know that we can pair each counting number with an integer in a one-to-one correspondence without missing any element.

counting numbers integers bijection

Using this concept, we show intuitively that the number of points on two line segments is equal even if they have different lengths. We can do this by showing that for each point on segment \overline{AB}, there is a corresponding point on segment \overline{CD}. » Read more

Counting the Real Numbers

If we are in a room full of ballroom dancers where each male dancer has a female dancer partner, and no one is left without a partner, we can say that there are as many male as female dancers in the room even without counting. In mathematics, we say that there is a one-to-one correspondence between the set of male dancers and the set of female dancers.

Pairing Infinite Sets

In the A Glimpse at Infinite Sets, we have learned that if we can pair two sets in one-to-one correspondence, we can say that the two sets have the same number of elements. The number of elements of a set is its cardinality. Therefore, the cardinality of the binary numbers {1,0} is 2 and the cardinality of the set of the vowel letters in the English alphabet {a, e, i, o, u} is 5.

The pairing of sets can be extended to compare sets with infinite number of elements or infinite sets.  In Figure 1, it is clear that it is possible to pair the set of integers with the set of counting numbers in one-to-one correspondence (can you see why?).  Infinite sets whose elements can be paired with the set of counting numbers in one-to-one correspondence is said to be countably infinite.

Figure 1

As a consequence of the analogy above, we can conclude the cardinality of counting numbers is equal to the cardinality of integers (Can you see why?). » Read more

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