Two standard ways of writing a set of numbers down: interval notation, which names the two ends and says whether each is included, and set-builder notation, which states the condition a number must satisfy. Both are exact, and both have a small number of forms that describe nothing.
Core ideas
The notation carries very little, and every part of it is load-bearing.
A square bracket includes the endpoint and a round one excludes it. [2, 5) contains 2 and does not contain 5. Mixing the two in one interval is ordinary rather than a mistake — it is exactly what at least 2 and less than 5 says.
∞ marks an end that never arrives. There is no number there to include, so the bracket beside it is always round: [2, ∞) and never [2, ∞]. The same holds for −∞ at the left end.
{ x ∈ ℝ : x ≥ 2 } reads the set of real numbers x such that x ≥ 2. It can express sets interval notation cannot — { x ∈ ℝ : x is rational } for one, which is not a union of finitely many intervals — and it makes explicit which numbers are being drawn from.
The material
Four ways to say the same thing, the pieces each notation is made of, and the handful of written forms that mean nothing.
Every answer in “Linear Inequalities”, “Compound Inequalities”, “Absolute Value Equations” and “Absolute Value Inequalities” was a set of numbers written as an inequality. That is a perfectly good way to write a set, and it becomes clumsy the moment the set has two pieces. It also says nothing about which numbers were on the table to begin with. Two standard notations address both.
For a ≤ b: [a, b] is the set of x with a ≤ x ≤ b; (a, b) is the set with a < x < b; [a, b) and (a, b] include one end each. A square bracket includes its endpoint and a round bracket excludes it.
Unbounded ends use ∞ and −∞, always with a round bracket: [a, ∞), (−∞, b], and (−∞, ∞) for the whole line.
{ x ∈ ℝ : P(x) } is the set of all real numbers x for which the condition P(x) is true. The colon is read such that, and a vertical bar is used for the same purpose: { x ∈ ℝ | x ≥ 2 }.
∈ is read is an element of. Naming the set the numbers come from matters: { x ∈ ℝ : 2 ≤ x ≤ 5 } is an interval, while { x ∈ ℤ : 2 ≤ x ≤ 5 } is the four numbers 2, 3, 4, 5.
The filled and hollow ends on the line are the brackets. A filled dot is a square bracket and a hollow one is round, and reading between the two representations should become automatic; the lab prints all four from one structure so that they cannot disagree.
x ≤ −2 or x ≥ 5 is two pieces, and interval notation writes it as a union: (−∞, −2] ∪ [5, ∞). That ∪ is “Compound Inequalities”'s or. The intersection sign ∩ appears far less often, because an and of two intervals is itself an interval and gets written as one.
The empty set is written ∅, or { }. It is not { 0 } and it is not 0: { 0 } is a set with one element in it, and ∅ is a set with none. An inequality with no solutions has solution set ∅, and writing 0 claims that 0 is a solution.
(2, 5) is also how a point in the plane is written, in Lines, Functions and Graphs. The two notations share their symbols and are told apart by context: an interval is a set of numbers on a line, and a point is a pair of coordinates. Where both are in play, saying which you mean is worth the words.
(−∞, −2] ∪ [5, ∞). Two pieces. The left one runs for ever downward and stops at −2, which it includes. The right one starts at 5, which it includes, and runs for ever upward. In set-builder notation it is { x ∈ ℝ : x ≤ −2 or x ≥ 5 }. Nothing strictly between −2 and 5 belongs to it, and both −2 and 5 do.
One further form is worth having. A set small enough to list is listed: { 5, −3 } is the solution set of 3|x − 1| + 2 = 14 from “Absolute Value Equations”, and the list is the readable form. Interval notation does reach it, as [−3, −3] ∪ [5, 5], and reaches it clumsily. Listing, interval notation and set-builder notation are three tools, and the set decides which of them is readable.
Interactive
Four notations for one set of numbers
Method
Four checks, in this order. The last one catches a bracket copied from the wrong side.
One piece is a single interval. Two or more are a union of intervals joined by ∪. A set small enough to list goes in braces. Nothing at all is ∅.
Interval notation is ordered. [−2, 5] is an interval and [5, −2] describes nothing, and the two are one transposition apart with no warning in between.
< and > give round brackets; ≤ and ≥ give square ones. The two ends are decided independently of each other, which is why [2, 5) is a perfectly ordinary interval.
Take the endpoint your notation claims is included and confirm that the original statement is true there. Then take a number from inside. A bracket written the wrong way round survives every other check and fails this one.
The solution sets from “Linear Inequalities”, “Compound Inequalities” and “Absolute Value Inequalities”, in each notation.
The bracket beside −6 is square because the relation that produced it was ≤. The bracket beside −∞ is round because there is nothing there to include, and that bracket is round in every interval unbounded on the left, without exception.
The third set is the one that needs the union. Written as an inequality it takes a word, or; written as an interval it takes a symbol, ∪. Both are exact, and the interval form is the one that scales: however many pieces a set has, it is a list of intervals joined by ∪.
For a faded two-piece conversion, start from x ≤ −3 or x > 2. The left piece has been started as (−∞, −3 and the right as (2, ∞). Choose the missing bracket at −3, join the pieces, write the real-number set-builder form, and describe the corresponding endpoint dots before checking the worked patterns above.
Translate (−∞, 1) ∪ [6, ∞) into an inequality, a real-number set-builder condition and a number line with correct endpoint dots. Then list { n ∈ ℤ : −2 < n ≤ 2 } and explain why the same condition over ℝ cannot be listed. Every later solution set is expected in one of these exact forms.
Something to change about this lesson? Write it down, tick it off when it is handled, and export the list when you want to act on it. Kept in this browser only: nothing is sent anywhere, and nobody is notified.
No recommendations yet for this lesson.