Intervals and Operations on Them

Mathematics — Second Preparatory

Worked examples

Example 1 Find the union, the intersection and the difference of the intervals A = [1, 5] and B = (3, 7].
Solution:

A ∪ B = [1, 7], because the first interval starts at 1 and the second ends at 7 and includes it.
A ∩ B = (3, 5], because the common numbers start just after 3 and end at 5, which is included.
A − B = [1, 3], because the numbers from 1 to 3 are in A, and 3 is not in B, so 3 is included.
Example 2 Solve the inequality 1 ≤ 2x − 3 < 7 and write the answer as an interval.
Solution Add 3 to every part to get 4 ≤ 2x < 10. Divide by 2 to get 2 ≤ x < 5. So the solution is the interval [2, 5).
Example 3 Find the union of the intervals [0, 2) and [2, 4].
Solution The first interval starts at 0 and does not include 2. The second starts at 2 and includes it, so there is no gap between them. Therefore [0, 2) ∪ [2, 4] = [0, 4].

Comparison aspect	Main idea	Keywords
Example 4	If (−∞ 4] represents the numbers less than or equal to 4 find the interval of the numbers common to it and (1	representsnumberslessequal
Solution	The common numbers are those greater than 1 and less than or equal to 4 that is (1 4]. Note that 1 is not in the second interval so the left endpoint is excluded	commonnumbersthosegreater

Intervals and Operations on Them · 5 / 7