# Definition:Roster Notation

## Definition

**Roster notation** is the technique of specifying the elements in a set by listing them between a pair of braces $\set{}$.

### Explicit Definition

A (finite) set can be defined by **explicitly** specifying *all* of its elements between the famous curly brackets, known as **set braces**: $\set {}$.

When a set is defined like this, note that *all* and *only* the elements in it are listed.

This is called **explicit (set) definition**.

It is possible for a set to contain other sets. For example:

- $S = \set {a, \set a }$

If there are many elements in a set, then it becomes tedious and impractical to list them all in one big long explicit definition. Fortunately, however, there are other techniques for listing sets.

### Implicit Definition

If the elements in a set have an *obvious* pattern to them, we can define the set **implicitly** by using an ellipsis ($\ldots$).

For example, suppose $S = \set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$.

A more compact way of defining this set is:

- $S = \set {1, 2, \ldots, 10}$

With this notation we are asked to suppose that the numbers count up uniformly, and we can read this definition as:

**$S$ is the set containing $1$, $2$, and so on, up to $10$.**