Sets: Real Number Line, Interval Notation & Set Notation

A set is a collection of “elements.”  The elements of a set can actually be anything: {January, February, March} is a set, but here we’ll be dealing primarily with real numbers.

The Real Number Line.

The most intuitive way to illustrate a set is by using the real number line.  If we draw a line, designate a point on the line to be zero, and choose a scale, then every point on the line corresponds uniquely to a real number:

The real number line respects the order of the real numbers.  A larger number will always be found to the right of a smaller number.  In the graphic above, π > –½ and –½ > –3.

We visualize a set on the real number line by marking its members.  It is standard to agree on the following conventions: To include an endpoint, we use a solid bubble.  To exclude an endpoint, we use an open bubble.

A bounded set will have both a beginning and an end point.  This bounded set represents all real numbers greater than –2 and less than or equal to 5.

An unbounded set omits one or both endpoints and indicates that the set continues infinitely in the unbounded direction.  The following unbounded set represents the set of all real numbers less than or equal to 4:

Sets do not need to be connected.  The this is the set all real numbers which are either greater than 3 or between –4 and –3:

The following depicts the set of all real numbers with the exception of –4 and 4:

Interval Notation.

Interval notation translates the information from the real number line into symbols.

The example  becomes the interval (-2,5].

To exclude an endpoint, we use parentheses; to include an endpoint, we use a bracket.

The example  is written in interval notation as (– , 4].

The infinity symbols +  and –  are used to indicate that the set is unbounded in the positive (+ ) or negative (– ) direction of the real number line.  +  and –  are not real numbers, therefore we may exclude them as endpoints by using parentheses.

If a set consists of disconnected elements, we use the symbol for union: .

The example  is written in interval notation as (–4,–3)  [3,+ ).

How do we express  in interval notation?

Here, we have three intervals to consider and describe the set as (– , –4)  (–4, 4)  (4, + ).

An interval where both endpoints are excluded, such as (–1, ), is called an open interval.  An interval is called closed, if it contains its endpoints, such as [–π]

An unbounded interval such as (–, 5) is considered to be open while an interval such as [π, + ] is called closed even though it does not contain its right endpoint.  The whole real line (– , + ) is considered to be both open and closed.

Set Notation.

The most flexible, and complicated, way to write down sets is to use set notation in which, sets are delimited by braces.  You can write down finite sets as lists, for instance, { –1, π } is the set with the three elements -1, π, and .

For sets with infinitely many elements this becomes impossible, so there are other ways to express them.  The symbol  denotes a set and special symbols are used for important sets:

  •  denotes the natural numbers.
  •  denotes the integers.
  •  denotes the set of rational numbers.
  •  denotes the set of all real numbers.

The interval (-3,5] is written in set notation as: {x  | −3 < x ≤ 5 } and is read as “the set of all real numbers x such that x is greater than −3 and less than or equal to 5.”  The first part tells us the realm of numbers we are describing.  We have defined the set as ” all real numbers x.”  The vertical bar represents the term “such that” and −3 < x ≤ 5 defines the interval.

The advantage of set notation is its flexibility.  Suppose we need a set with an interval ≥−38 and ≤+130 but we only want the integers.  Using set notation we can accurately describe the set as {x  | −38 ≤ x ≤ 130 } which reads as “the set of all integers x such that x is greater than or equal to −38 and less than or equal to +130.”

The set {x  | −3 < x ≤ 5 } is the set of all integers greater than −3 and not greater than 5.  Let’s compare it with {x  | −3 < x ≤ 5 }

Both have an interval of (−3, 5] but the first set calls for the integers {−2, −1, 0, 1, 2, 3, 4, 5} whereas the second specifies natural numbers and would include only {1, 2, 3, 4, 5}.

Here are some more examples:

The interval (3, ) is written as {x  | x > 3 }

The set (−, 0)  (0, +) can be written as {x  | x ≠ 0 } or as {x  | x < 0 or x > 0 }

Exercise 1

Write the set of all real numbers strictly between -2 and π in interval notation and in set notation.

Exercise 2

Write the set (−, 1]  (3, ) in set notation.

Exercise 3

Write the unbounded set:  in both interval notation and set notation.

Exercise 4

Mark the set {x  |  < x ≤  } on a real number line.

Exercise 5

Solve the inequality, 2 − x ≤ x + 3, and define its set in all three notations.

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