Cardinality of a Set

The cardinality of a set S, denoted by |S|, is the number of elements in the set. This number is also referred to as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞.

Examples of Cardinality

|{1, 4, 3, 5}| = 4          (finite set with 4 elements)

|{1, 2, 3, 4, 5, ...}| = ?  (infinite set of natural numbers)

|{}| = 0                     (empty set has cardinality 0)

Comparing Cardinalities of Two Sets

If there are two sets X and Y, their cardinalities can be compared as follows −

• |X| = |Y| − Sets X and Y have the same cardinality. This occurs when the number of elements in X is exactly equal to the number of elements in Y. In this case, there exists a bijective function f from X to Y (a one-to-one correspondence).
• |X| ≤ |Y| − Set X's cardinality is less than or equal to set Y's cardinality. This occurs when the number of elements in X is less than or equal to that of Y. Here, there exists an injective function f from X to Y.
• |X| < |Y| − Set X's cardinality is strictly less than set Y's cardinality. The function f from X to Y is injective but not bijective.
• If |X| ≤ |Y| and |X| ≥ |Y|, then |X| = |Y| − The sets X and Y are commonly referred to as equivalent sets. This is known as the Cantor–Bernstein theorem.

Example

Consider the following sets −

X = {a, b, c}       |X| = 3
Y = {1, 2, 3}       |Y| = 3
Z = {p, q, r, s}    |Z| = 4

|X| = |Y|    ? X and Y have the same cardinality (equivalent sets)
|X| < |Z|    ? X has strictly fewer elements than Z
|X| ? |Z|    ? X's cardinality is less than or equal to Z's

Since |X| = |Y| = 3, we can define a bijective function f: X → Y such as f(a) = 1, f(b) = 2, f(c) = 3. This confirms the two sets have equal cardinality.

Conclusion

Cardinality measures the size of a set by counting its elements. Two sets have equal cardinality when a bijective function exists between them, and cardinality comparisons are established through the existence of injective or bijective functions.