📂Set Theory

Cardinality of a Set

Definition ¹

For any given set $X$, $\operatorname{card} X$ that satisfies the following properties is defined as the Cardinality of $X$.

• (i): $X = \emptyset \iff \operatorname{card} X = 0$
• (ii): $A \sim B \iff \operatorname{card} A = \operatorname{card} B$
• (iii): For some natural number $k$, if $X \sim \left\{ 1 , 2, \cdots , k \right\}$ then $\operatorname{card} X = k$

Specifically, the cardinality of a finite set is called a finite cardinality, and that of an infinite set is called a transfinite cardinality.

1. For two sets $A$ and $B$, if $A$ is equivalent to some subset of $B$ but $B$ is not equivalent to any subset of $A$, then it is said that $\operatorname{card} A$ is smaller than $\operatorname{card} B$, which is represented as follows: $$\operatorname{card} A < \operatorname{card} B$$
2. For two disjoint sets $A$ and $B$, each having cardinalities $a = \operatorname{card} A$ and $b =\operatorname{card} B$ respectively, the cardinality of their union is called the sum of the cardinalities $a$ and $b$, and is represented as follows: $$\operatorname{card} \left( A \cup B \right):= a+b$$
3. For two sets $A$ and $B$, each having cardinalities $a = \operatorname{card} A$ and $b =\operatorname{card} B$ respectively, the cardinality of their Cartesian product is called the product of the cardinalities $a$ and $b$, and is represented as follows: $$\operatorname{card} \left( A \times B \right):= ab$$
4. For two sets $A$ and $B$, each having cardinalities $a = \operatorname{card} A$ and $b =\operatorname{card} B$ respectively, the cardinality of the set of all functions with domain $A$ and codomain $B$ $B^{A}$ is called the $b$ to the power of $a$ (cardinality), and is represented as follows: $$\operatorname{card} \left( B^{A} \right):= b^{a}$$

Explanation

Cardinality is an ‘abstraction of the size of a set’, and it is reasonable to say that it was introduced to make mathematically meaningful comparisons for infinite sets as well. Since it comes from the concept of the size of a set, it is often represented simply as $|X| := \operatorname{card} X$ when set theory is not the core or for convenience.

Cardinality has the following algebraic properties similar to natural numbers.

Basic Properties ¹

Let’s say $x,y,z$ is a cardinality.

• [1]: $$|A| \le |B| \land |A| \ge |B| \implies |A| = |B|$$
• [2]: $$x + y = y+x \\ (x+y) + z = x + (y + z)$$
• [3]: $$xy = yx \\ (xy)z = x(yz) \\ x ( y+z) = xy + xz$$
• [4]: $$z^{x} z^{y} = z^{x+y} \\ \left( z^{y} \right)^{x} = z^{yx}$$