1. What is an interval?

An interval is a subset of the real numbers written between brackets. When writing an interval, the smaller number is written first.

Note : A square bracket [ or ] means "included" ($\leq$ , $\geq$). A round bracket ( , ) written as ] , [ means "not included" (< , >).

2. Special intervals

• R = ]-$\infty$ , $\infty$[ (the set of all real numbers)
• R⁺ = ]0 , $\infty$[ (positive real numbers)
• R⁻ = ]-$\infty$ , 0[ (negative real numbers)
• R = R⁺ $\cup$ {0} $\cup$ R⁻
• Non-negative reals = R⁺ $\cup$ {0} = [0 , $\infty$[
• Non-positive reals = R⁻ $\cup$ {0} = ]-$\infty$ , 0]

3. Operations on sets (recall)

If X = {1 , 2 , 3 , 4} , Y = {3 , 4 , 5 , 6} :

• Union X $\cup$ Y = {1 , 2 , 3 , 4 , 5 , 6} (all elements in X or Y)
• Intersection X $\cap$ Y = {3 , 4} (elements in both)
• Difference X - Y = {1 , 2} (elements in X but not in Y)
• Complement X' = U - X (elements of the universal set U not in X)

4. Operations on intervals

Intervals are subsets of R, so we can perform the same operations using the number line.

Rule of thumb for interval results:

• Union ( $\cup$ ): the wider interval that covers both, including any gap between them.
• Intersection ( $\cap$ ): the overlapping part of the two intervals (maybe empty $\emptyset$).
• Difference ( - ): the part of the first interval not covered by the second.
• Complement ( ' ): R minus the interval (for intervals written with $\infty$).

Example : x = ]-2 , 2[ , y = [1 , 4]

• x $\cup$ y = ]-2 , 4]
• x $\cap$ y = [1 , 2[
• x - y = ]-2 , 1[
• y - x = [2 , 4]

5. Mixing intervals with sets (points)

• A single number (point) like {3} : the union of an open interval with its endpoint closes the bracket.

Example : ]2 , 5[ $\cup$ {2 , 5} = [2 , 5]

• Example : [3 , 5] $\cup$ {3 , 5} = [3 , 5] (already included).
• Example : ]3 , 5[ $\cap$ {3 , 5} = $\emptyset$ (neither 3 nor 5 is inside the open interval).
• Example : [2 , 5] - {2 , 5} = ]2 , 5[.

6. REMEMBER

• The complement of an interval uses the universal set R : x = ]-$\infty$ , 4[ $\rightarrow$ x' = [4 , $\infty$[.
• N $\cap$ [1 , 4] = {1 , 2 , 3 , 4}.

7. WORKED EXAMPLES

Example 1 : Using the number line, if x = [2 , 5[ and y = [-1 , 3[ find x $\cap$ y.

• The overlap of [2 , 5[ and [-1 , 3[ is [2 , 3[. Answer : x $\cap$ y = [2 , 3[.

Example 2 : x = [1 , 3] , y = ]0 , 2] find x $\cup$ y.

• Union covers from 0 (excluded) to 3 (included) $\rightarrow$ x $\cup$ y = ]0 , 3].

Example 3 : Write { x : x $\in$ R , x $\geq$ -2 } as an interval.

• Answer : [-2 , $\infty$[.