Sets: A Gentle Introduction

There are people who are fond of buying things in sets. For example, a pen collector may collect pens of different colors.   Other people are obsessed with brands.   For instance, Apple fans may buy the following an iPhone, iPod, iPad, and an iMac. If you are fond of reading Dan Brown, you probably have read or own some of the following books: The Lost SymbolDeception PointDigital Fortress, and The Da Vinci Code, and others.

Pens

In mathematics, a set is not very different.  A set is can be described as a collection of objects with common characteristics. These objects are called elementsGiven a set and an object, it is clear if the object is an element of a set or not.  If we let D be a set of Dan Brown novels, then D = { The Lost SymbolDeception PointDigital Fortress, The Da Vinci Code}. Also, if we let V be the set of vowel letters in the English alphabet, then V = {a, e, i, o, u}, and if we let R be the set of polygons whose number sides is less than 9, then the set would include the polygons in the figure below. Continue reading

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?). Continue reading

Counting the Uncountable: A Glimpse at Infinite Sets

When counting, we pair the counting numbers (positive integers) with the objects that we are counting.  For example, when we count our fingers, we assign a corresponding number to each finger.  A possible assignment would be 1 to the pinkie, 2 to the ring finger, 3 to the middle finger, 4 to the index finger, and 5 to the thumb. Of course, the order does not matter as long as we have a one-to-one pairing. As we can see, counting is easy once the objects that we are counting is finite.

Higher mathematics, however, does not just deal with finite number of objects. Sometimes, mathematicians need to examine sets of objects with infinite number of elements.  Such sets are called infinite sets. The set of integers Z and the set of counting numbers N are examples of infinite sets.

One strategy used  to examine the number of elements in an infinite set is to find another infinite set that is easier to enumerate, then compare them.  If a strategy can be done such that each element in the first set can be paired with exactly one element in the second set without missing anything, then it follows that the two sets have the same number of elements.

This may sound a little absurd at first, but let us have the following analogy:  If we want to know how many pairs of shoes are available in a shoe store, we do not have to count all the shoes.  We can just count the number of left-foot shoes, and we will know how many shoes are there. Our assumption, of course, is that every shoe has a pair.

Table 1 – Counting Numbers and Integers Pairing

With the abovementioned strategy, let us try to compare the set of counting numbers N and the set of the integers Z.  In Table 1, it is clear each element in N can be paired with exactly one element in Z, such that N is in increasing order and Z in alternating positive an negative signs. As we  can see, this strategy will not miss any integer as N increases without bound.   It is also apparent that even if the pairing continues forever, we are sure that each element in N has a pair in Z. Hence, the number positive integers N is the same as the number of all the integers Z including 0 and negative!

Surprised?

This is also the same with rational numbers. The number of elements in the set of positive rational numbers Q is the same as the number elements in the set of counting numbers N.  In Table 2, we can see the first 15 pairs of the one-to-one pairing.  There are duplicates such as 1/2 and 2/4, but we can easily eliminate equivalent fractions and replace them with the next rational numbers.  From the table, it is easy to see that we can devise a way to pair the set N (red texts) with the set Q (yellow texts) without missing anything.

Can you see the pattern?

Therefore, we can conclude that the number of positive rational numbers are as many as the number of counting numbers.

Table 2 – Counting numbers and rational numbers pairing.

From our discussion above, we can conclude that the set of counting numbers N, the set of integers Z, and the set of rational numbers Q have the same number of elements.

Going Technical

In set theory, the number of elements of a set is called cardinality.  Hence the cardinality of the set of vowel letters in the English alphabet {A, E, I, O, U} is 5, and the cardinality of the set of binary numbers {1,0} is 2.

Venn Diagram of the set of N, Z and Q.

It is interesting to note that there is a little bit of irony in our discussion above. We know that N is a subset of Z, and Z is a subset of Q, yet the three sets have the same cardinality.

The symbol \aleph_0  (or aleph null) is the symbol assigned to the cardinality of counting numbers.

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