BioMath: Mathematical Notation

The Biology Project > Biomath > Notation > Set-Builder and Interval Notations

Mathematical Notation

Set-Builder and Interval Notations

Set-Builder Notation

Set-builder notation is commonly used to compactly represent a set of numbers. We can use set-builder notation to express the domain or range of a function. For example, the set given by,

{x | x ≠ 0},

is in set-builder notation. This set is read as,

“The set of all real numbers x, such that x is not equal to 0,”

(where the symbol | is read as such that). That is, this set contains all real numbers except zero.

Symbol	Represents
{ }	Denotes the set
\|	Such that

Another example of set-builder notation is,

{x | − 2 < x ≤ 3} .

This set is read as,

“The set of all real numbers x, such that x is greater than −2 and less than or equal to 3.”

As stated above, we can use set-builder notation to express the domain of a function. For example, the function

has domain that consists of all real numbers greater than or equal to zero, because the square root of a negative number is not a real number. We can write the domain of f(x) in set builder notation as,

{x | x ≥ 0}.

If the domain of a function is all real numbers (i.e. there are no restrictions on x), you can simply state the domain as, ‘all real numbers,’ or use the symbol to represent all real numbers.

Interval Notation

We can also use interval notation to express the domain of a function. Interval notation uses the following symbols

Symbol	Represents
∪	Union of two sets
( )	An open interval (i.e. we do not include the endpoint(s))
[ ]	A closed interval (i.e. we do include the endpoint(s))

Interval notation can be used to express a variety of different sets of numbers. Here are a few common examples.

A set including all real numbers except a single number.

The union symbol can be used for disjoint sets. For example, we can express the set,

{x | x ≠ 0},

using interval notation as,

(−∞, 0) ∪ (0, ∞).

We use the union symbol (∪) between these two intervals because we are removing the point x = 0.

We can visualize the above union of intervals using a number line as,

Number line depicting all real numbers except zero

Notice that on our number line, an open dot indicates exclusion of a point, a closed dot indicates inclusion of a point, and an arrow indicates extension to −∞ or ∞.

Open and closed intervals

Now let's look at another example. The set given by,

{x| − 2 < x ≤ 3} ,

can be expressed in interval notation as,

(−2, 3].

We can visualize this interval using a number line as,

Number line with open dot on -2, closed dot on 3, and including all numbers from -2 to 3

A set including all real numbers

If the domain of a function is all real numbers, you can represent this using interval notation as (−∞,∞).

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In the next section we will describe summation notation.

Summation Notation

The Biology Project > Biomath > Notation > Set-Builder and Interval Notations

The Biology Project
Department of Biochemistry and Molecular Biophysics
The University of Arizona
January 2006
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