Definition:Roster Notation

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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 $10$.