SetBuilder Notation
Setbuilder notation is commonly used to compactly represent a set of numbers. We can use setbuilder notation to express the domain or range of a function. For example, the set given by,
{x  x ≠ 0},
is in setbuilder 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 setbuilder 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 setbuilder 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,
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,
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 