MAT 344: Cardinality

Question

Can you please remind me what is the meaning of "cardinality", i.e, card(S), where S is some set?

Answer

The cardinality of a finite set is the number of members that set has. The set {'A', 'B', 'C'} has cardinality three, the empty set has cardinality zero. The word 'cardinality' comes originally from the term 'cardinal number', meaning a number like one, two or three, as opposed to an 'ordinal number', meaning a number like first, second or third. ('Cardinal' itself comes from a Latin word meaning 'hinge', as in 'basic, important things on which everything hangs', and is etymologically the same word that is used in English to describe a high-ranking Catholic priest, or a bright red bird).

Infinite sets can have cardinalities too. It is not too difficult to show that the set of integers and the set of rational numbers have the same cardinality. (To do this, find a mapping between the two that is 1-1 and onto.) This cardinality is called aleph-null, and it's also not hard to show that it is less than the cardinality of the set of real numbers. (What do you need to do to prove this? It's not as easy as you might think.)

There are also infinite ordinals, and depending on how you define them, there are more of them than there are infinite cardinalities, but this is beginning to get beyond the scope of a combinatorics course.