Which method describe a set by listing down all the elements contained in the set?

A set can be described two ways -- by roster method or by using set-builder notation.

Inhaltsverzeichnis Show

Section 5.2 Roster Form
Definition5.2.1.
Example5.2.3.Sets in roster form.
Checkpoint5.2.4. Write the set in roster form.
Example5.2.5.Sets in roster from with ellipsis.
Checkpoint5.2.6.Write the set in roster form.
Definition5.2.7.
Which method describes a set by listing down all the elements contained in the set?
What method of describing sets lists all the elements?
Is a method of describing a set which involves of listing the elements of the set enclosing them by braces and separating each of them by a comma?

1. Roster Method

The roster method simply lists all the elements in the set. For example, set A could be described using braces like this:

A = {1, 2, 3, 4, 5}.

We could easily tell what is in the set by just looking at it. The above is a finite set, meaning you can count the number of objects in the set.

Another version of the roster method that is used when dealing with infinite set is sometimes called the modified roster method because it uses an ellipsis " … " to show that it follows a given pattern to infinity. For example, B = {1, 2, 3, … } would be the infinite set of all positive integers.

The modified roster method can also be used on finite sets where there are many terms that follow a pattern. For example, C = {1, 2, 3,…100} would be the set of positive integers from one to one hundred. Here, the three dots show that the numbers follow the pattern set up by the first three numbers to one hundred.

Note: In set B above, we cannot list all the numbers and we cannot count them, so set B is an infinite set.

If a set is described using the roster method, you must use braces to enclose the elements.

2. Set-Builder Notation

The second method of describing a set is set-builder notation, wherea set is described using the following format:

A = {x | x is an even integer > 0}

It would be read as A is the set of all x such that x is an even integer greater than zero. This set would look like the following using the roster method:

A = {2, 4, 6, …}.

The set-builder notation method can also be used to describe a set that would be cumbersome using the roster method. For example,

A = { x | x is a city in the U.S. that has more than 5,000 people living in it}.

You can imagine how many cities would be in this set!

Empty Set

Now that we know what a set is and can describe it, we need to talk about a very special set called the empty set or the null set. The more common name is empty set. It is defined as the set containing no elements and is denoted by the symbol Ø or by empty braces { }.

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Section 5.2 Roster Form

So far we specified the elements of sets by verbally. The roster form introduced here offers a concise way of writing down sets by listing all elements of the set. Furthermore we use ellipsis to describe the elements in a set, when we believe that the reader understands how a pattern in a list of elements continues.

Definition5.2.1.

The contents of a set can be described by listing the elements of the set, separated by commas, inside a set of curly brackets. This way of describing a set is called roster form.

In the video in Figure 5.2.2 we recall the definition of roster form and give first examples.

Figure5.2.2.Roster Form by Matt Farmer and Stephen Steward

We give examples of sets in roster form along with a verbal description.

Example5.2.3.Sets in roster form.

\(\{1, 2, 3, 4\}\) is the set containing the numbers 1, 2, 3, and 4.
\(\{\mathtt{w}, \mathtt{x}, \mathtt{y}, \mathtt{z}\}\) is the set containing the letters \(\mathtt{w}, \mathtt{x}, \mathtt{y}\text{,}\) and \(\mathtt{z}\text{.}\)
\(\{ \text{red}, \text{yellow}, \text{blue} \}\) is the set containing red, yellow, and blue.
\(\{6\}\) is the set containing the number \(6\text{.}\)
\(\{3,-3,11\}\) is the set containing the numbers \(3\text{,}\) \(-3\text{,}\) and \(11\text{.}\)
\(\{5,3,\mathtt{w}\}\) is the set containing the numbers \(5\) and \(3\) and the letter \(\mathtt{w}\text{.}\)

Now try yourself to translate a verbal description of a set into roster form.

Checkpoint5.2.4. Write the set in roster form.

Give the set of natural numbers less than \(7\) in roster form:

\(\lbrace\)\(\rbrace\)

Recall that an ellipsis (\(\ldots\)) indicates that the pattern is continued. We can use an ellipsis when writing a set in roster form instead of listing every element.

Example5.2.5.Sets in roster from with ellipsis.

We give examples of sets written in roster form that use ellipses.

\(\{1,2,3,\ldots,100\}\) is the set of integers from \(1\) to \(100\text{.}\)
\(\{2,3,4,\ldots,99\}\) is the set of integers from \(2\) to \(99\text{.}\)
\(\{\mathtt{c},\mathtt{d},\mathtt{e},\ldots,\mathtt{n}\}\) is the set of letters from \(\mathtt{c}\) to \(\mathtt{n}\text{.}\)

Convert a set given in roster form with ellipsis into roster form without ellipsis.

Checkpoint5.2.6.Write the set in roster form.

Give the set in roster form (without ellipsis):

\(\lbrace8,9,10,...,16\rbrace = \lbrace\)\(\rbrace\)

Answer.

\(8, 9, 10, 11, 12, 13, 14, 15, 16\)

Roster form also allows us to formulate a set that does not contain any elements by writing \(\{ \}\text{.}\)

Definition5.2.7.

The empty set (also called the null set) is the set that consists of no elements. It is denoted by \(\{ \}\text{.}\)

The empty set contains no elements. In particular it does not contain the number \(0\text{.}\)

Which method describes a set by listing down all the elements contained in the set?

Listing The Elements (Roster Method) The set can be defined by listing all its elements, separated by commas and enclosed within braces. This is called the roster method.

What method of describing sets lists all the elements?

The roster method lists all the elements or members in the set, whereas a description in words explains what elements are in the set using a sentence.