Advanced Mathematics

2.4Cardinality of Sets

Purpose of Section

To introduce the concept of cardinality of a set and the equivalence of two sets. We also define what it means for a set to be finite and infinite and how to compare sizes of sets, both finite and infinite. We end our discussion by defining the cardinality of the natural numbers.

2.4.1 Introduction

No one knows exactly when people first started counting, but a good guess might be when people started accumulating things. However, long before number systems were invented, two people might have determined whether they had the same number of goats and sheep as illustrated in Figure 2.20 by simply placing them in a one‐to‐one correspondence with each other.

Figure 2.20 Counting sheep.

Another person might have designated a stone for each goat, thus obtaining a one‐to‐one correspondence between the goats and a pile of stones. Today, we no longer use stones to enumerate things since we have symbolic stones in the form of 1, 2, …. To determine the number of goats, we simply “count,” 1, 2, … and envision the rocks R = {1, 2, 3, 4, 5} in our mind as illustrated in Figure 2.21.

Figure 2.21 Modern way to count.

Throughout the history of mathematics, the subject of infinity has been mostly taboo, more apt to be included in discussions ...