Set-Builder Notation

Section 5.5 Set-Builder Notation

Set-builder notation can be used to specify a set by describing the properties of its elements. In set-builder notation we write sets in the form

{x ∣ (properties of x)},

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where (properties of

x

) is replaced by conditions that fully describe the elements of the set. The bar (

∣

) is used to separate the elements and properties. The bar is read as “such that,” and all together we read this set as “the set of all elements

x

such that (properties of

x

).” We use a variable (here

x

) to formulate the properties on the elements in the set.

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With a concrete example we illustrate, how set builder is read and interpreted.

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Example 5.37. Reading set-builder notation.

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We consider the set

A = {x ∣ x \in N and x < 6} .

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We read this as

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$A$ is equal to the set of all elements $x$ such that $x$ is a natural number and $x$ is less than $6 .$

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Notice how we use

x

to formulate the properties of the elements in the set. Knowing what the elements of the set of natural numbers

N

and that the set

A

only contains numbers that are less than

6

we find that the set

A

written in roster form is

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

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In the example above two properties of the elements in the set were given. We now consider a set defined by four properties of its elements.

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Problem 5.38. Four properties.

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Give the set in roster form.

{x ∣ x \in Z and x > 12 and x < 20 and x mod 3 = 1}

Solution.

We go through the conditions on the elements of the set one by one.

The first property is

x \in Z,

so the possible values for

x

are:

\dots, - 2, - 1, 0, 1, 0, 1, 2, \dots

The first two properties are

x \in Z

and

x > 12,

so the possible values for

x

are:

13, 14, 15, 16, 17, \dots

The first three properties are

x \in Z

and

x > 12

and

x < 20,

so the possible values for

x

are:

13, 14, 15, 16, 17, 18, 19

Now we find the elements that satisfy all four properties

x \in Z

and

x > 12

and

x < 20

and

x mod 3 = 1,

that is we find all values of

x

from the list above that satisfy

x mod 3 = 1 :

\begin{matrix} 13 mod 3 = 1 \\ 14 mod 3 = 2 \\ 15 mod 3 = 0 \\ 16 mod 3 = 1 \\ 17 mod 3 = 2 \\ 18 mod 3 = 0 \\ 19 mod 3 = 1 \end{matrix}

So the elements satisfying all four properties are

13

and

16

and

19 .

Thus

{x ∣ x \in Z and x > 12 and x < 20 and x mod 3 = 1} = {13, 16, 19} .

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In the video in Figure 5.39 we recall the definition of set-builder notation and give examples of sets written in set-builder notation.

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Figure 5.39. Set-Builder Notation by Matt Farmer and Stephen Steward

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There are several ways of representing the same set. We can describe the same set verbally, in roster form, or in roster form with ellipsis. Set-builder notation yields even more ways of representing the same set.

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Example 5.40. A set written in three ways.

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There are many ways of describing the same set using set-builder notation:

${x ∣ x is a natural number from 4 to 8} = {4, 5, 6, 7, 8}$
${x ∣ x \in N and x > 3 and x < 9} = {4, 5, 6, 7, 8}$
${x ∣ x \in N and x \geq 4 and x \leq 8} = {4, 5, 6, 7, 8}$

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Many sets that we have encountered before can also be formulated in set builder notation.

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Example 5.41. Selected sets in set-builder notation.

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We formulate some familiar sets in set-builder notation.

${x ∣ x \in Z and x > 0}$ is the set of positive integers, also known as the set of natural numbers.
${x ∣ x \in N and x mod 2 = 0} = {2, 4, 6, 8, \dots}$ is the set of even natural numbers.
${x ∣ x \in N and x mod 2 = 1} = {1, 3, 5, 7, \dots}$ is the set of odd natural numbers.
${x ∣ x \in N and x < 0} = {}$ as there are no natural numbers that are less than zero.

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We represent some special sets in set-builder notation.

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Example 5.42. Special sets in set-builder notation.

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We give special sets from the previous section in set builder notation. Let

m \in N .

$Z_{m} = {x ∣ x \in Z and x \geq 0 and x < m} .$
$Z_{m}^{\otimes} = {x ∣ x \in Z and x > 0 and x < m} .$

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Now read a set in set-builder notation, formulate a verbal description and give the set in roster form.

Checkpoint 5.43. Read set-builder notation, write in roster form.

Consider:

{t ∣ t \in Z and t > 30 and t < 35}

This is read as:

The set

select
of the one element
of all elements

t

select
where
with the exception that

t

select
an element of
not an element of
less than
greater than
equal to

the set of

select
natural numbers
integers
negative integers
characters

and

t

select
less than
less than or equal to
greater than
greater than or equal to
equal to
not equal to
better than
worse than

30

and

t

select
less than
less than or equal to
greater than
greater than or equal to
equal to
not equal to
better than
worse than

35 .

Give the set in roster form.

{

}

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