Finite and Infinite Sets

The Idea of "How Many Elements?"

When we look at a set, the first natural question is: how many distinct objects does it contain? For a set like $A = \{1, 2, 3, 4, 5\}$, the answer is straightforward — five elements. For $B = \{a, b, c, d, e, g\}$, it is six. But what about the set $C = \{\text{men living presently in different parts of the world}\}$? We cannot count them instantly, but we know the number is some natural number — possibly very large, but still a definite, finite number.

This idea of "the number of distinct elements" is so fundamental that we give it a notation. For any set $S$, we write $n(S)$ to denote the number of distinct elements in $S$. So $n(A) = 5$, $n(B) = 6$, and $n(C)$ is some finite natural number.

Now consider the set of natural numbers, $\mathbb{N} = \{1, 2, 3, 4, \dots\}$. Can we assign a finite number to $n(\mathbb{N})$? No — the list never ends. There is no largest natural number; you can always add one more. Such a set is called infinite.

Finite Sets — The Definition

A set is called finite if it is either empty or consists of a definite number of elements. In other words, if $n(S)$ is a natural number (including zero for the empty set), then $S$ is a finite set.

A set is called infinite if it is not finite.

Let's test this with examples:

• The set $W$ of the days of the week: $W = \{\text{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}\}$. $n(W) = 7$, a natural number. So $W$ is finite.
• The set $S$ of solutions to $x^2 - 16 = 0$: solving gives $x = 4$ or $x = -4$, so $S = \{4, -4\}$. $n(S) = 2$, finite.
• The set $G$ of all points on a line: between any two distinct points there are infinitely many other points. You cannot count them. $G$ is infinite.

Representing Infinite Sets in Roster Form

When we write a set in roster form, we list all elements inside curly braces. For a finite set, this is possible. For an infinite set, we cannot list every element — there are too many. So we use a shorthand: we write a few elements that clearly show the pattern, followed by three dots (ellipsis).

Examples:

• The set of natural numbers: $\{1, 2, 3, \dots\}$ …