Week 2 · Intervals: bounded and unbounded intervals and the operations union, intersection, difference and complement · 3 lessons
1
Bounded intervals: closed, open and half-open on the number line
Mathematics Prep 2 Term 1, Unit 1, Lesson 2
GoalWrite a bounded interval as an inequality, in interval notation and on the number line, and decide which end points belong to it.
New words
interval · all the real numbers between two end pointsclosed interval [a, b] · includes both end points: a ≤ x ≤ bopen interval ]a, b[ · leaves out both end points: a < x < bhalf-open interval · includes one end point only: [a, b[ or ]a, b]
Explanation
An interval is a set of real numbers lying between two end points. It can be written in three ways: as a set {x : x ∈ R, 2 ≤ x < 5}, in interval notation [2, 5[, and on the number line. A square bracket next to a number (or a filled dot) means the number is included, which goes with ≤ or ≥. A bracket pointing outwards (or an empty circle) means the number is left out, which goes with < or >. So [a, b] is a ≤ x ≤ b, ]a, b[ is a < x < b, ]a, b] is a < x ≤ b and [a, b[ is a ≤ x < b. An open end really is left out: in ]2, 5[ the number 5 is missing, but 4.999 is in, and there is no largest number below 5.
Name
Interval
Inequality
closed
[2, 5]
2 ≤ x ≤ 5
open
]2, 5[
2 < x < 5
half-open
]2, 5]
2 < x ≤ 5
half-open
[2, 5[
2 ≤ x < 5
Worked examples
Describe [2, 5]. It is {x : x ∈ R, 2 ≤ x ≤ 5}. Both ends are included, so 2 ∈ [2, 5] and 5 ∈ [2, 5]. Also √5 ≈ 2.24 is in it, but 1.9 is not.
Describe ]−1, 4[. It is {x : x ∈ R, −1 < x < 4}. Neither −1 nor 4 belongs to it: −1 ∉ ]−1, 4[ and 4 ∉ ]−1, 4[. But 0 and 3.99 do belong.
Read the number line. The empty circle is at −2 and the filled dot at 3, so the interval is ]−2, 3] = {x : x ∈ R, −2 < x ≤ 3}.
Class activity
Human number line. With chalk, draw a line from 0 to 10 on the ground. Choose the interval ]2, 7]. Ask friends to stand on the numbers 0, 1, 2, …, 10 that belong to it. Who stands on 2? Who stands on 7?
Practice
1
Write ]3, 7] as an inequality.
3 < x ≤ 7.
2
Write {x : x ∈ R, −2 ≤ x < 4} as an interval.
[−2, 4[.
3
Decide: 5 ∈ ]1, 5[ ? √10 ∈ ]3, 4] ?
5 ∉ ]1, 5[ because the end is open. √10 ≈ 3.16 is between 3 and 4, so √10 ∈ ]3, 4].
4
Write the interval shown on the number line.
[1, 6[ (filled dot at 1, empty circle at 6).
5
Karim writes ]2, 5] for 2 ≤ x < 5. What is wrong?
The brackets are the wrong way round. 2 is included, so the left bracket is square; 5 is left out, so the right one is open: [2, 5[.
