You already know that sets can be described by listing their elements. But often, it's more powerful to define a set by the rule its members must follow. This is called set-builder notation. The real skill is translating this rule-based format into a simple list, also known as Roster form .

Let's take an example. Consider the set A, defined as:

A = {x : x is a positive integer and $x^2 < 40$}

Here, 'x' is our variable. The colon means 'such that'. The rule is that x must be a positive integer whose square is less than 40. To convert this to Roster form, we just test the numbers.

$1^2 = 1 (< 40)$ $2^2 = 4 (< 40)$ $3^2 = 9 (< 40)$ $4^2 = 16 (< 40)$ $5^2 = 25 (< 40)$ $6^2 = 36 (< 40)$ $7^2 = 49 (> 40)$ - too high.

So, the Roster form of set A is {1, 2, 3, 4, 5, 6}.

Things get interesting when we talk about subsets of the Real numbers (denoted by the symbol R). It's impossible to list all the numbers between 1 and 2—there are infinitely many! Instead, we use a special notation called intervals.

An interval is a continuous section of the real number line. We define it by its endpoints. There are four main types, depending on whether the endpoints themselves are included in the set.

Let's break down these notations. Suppose we have two real numbers, a and b, where $a < b$.

The parenthesis () means the endpoint is not included. The square bracket [] means it is included.

So, if we see the set-builder notation {x ∈ R : -5 ≤ x < 2}, we can immediately write it in interval notation as [-5, 2). This concise format is extremely common when defining the domain of functions.

A simple but important concept is the length of an interval. For any of the four interval types between endpoints a and b, the length is the same.

For example, the length of the interval [-3, 10] is $10 - (-3) = 13$. The length of the open interval (-3, 10) is also 13. This might seem counterintuitive, but because there's no 'next' real number after -3, the inclusion of the single point endpoint doesn't add to the overall length of the span.

Being able to switch between these notations is a key skill for your exams.