Types of Sets — Empty, Finite, Infinite and Equal Sets

Sets can be classified by how many elements they contain, and we can also ask when two

sets, described differently, are actually the same set.

The empty set. A set that contains no elements at all is called the empty set

(or null set), denoted $\varnothing$ or $\{\ \}$. For instance,

because $2x + 1 = 0$ gives $x = -\tfrac{1}{2}$, which is not a natural number — so no

natural number satisfies the condition, and the set has no elements. Note that

$\{0\}$ and $\{\varnothing\}$ are not the empty set: each is a set containing one

element (the number $0$, or the empty set itself, respectively).

Finite and infinite sets. A set is called a finite set if it is either empty

or its elements can be counted by a definite (finite) whole number — that is, the

counting process of listing its elements terminates. A set that is not finite is

called an infinite set. For example, the set of letters of the English alphabet is

finite ($n = 26$), while $\{x \in N : x \text{ is a multiple of } 5\}$ is infinite,

because however many multiples of 5 we list, there is always a larger one still to be

listed. The set of all lines parallel to a given line is also infinite, since through

every point not on the given line there passes exactly one such parallel line, and

there are infinitely many such points.

Equal sets. Two sets $A$ and $B$ are said to be equal, written $A = B$, if

every element of $A$ is an element of $B$ and every element of $B$ is an element of

$A$ — that is, they contain exactly the same elements. The order in which elements are

listed, and any repetition, is irrelevant. For example,

since all three descriptions name the same three distinct objects $1, 2, 3$. If even

one element differs between two sets, they are not equal — for example …