It seems like you're asking about intervals in mathematics. Could you clarify if you mean intervals in terms of numbers, functions, or something else? I'll give you an overview of intervals in real numbers to start:

In mathematics, intervals represent a range of numbers on the real number line. They specify the set of all numbers between two endpoints, and the endpoints can be included or excluded depending on the type of interval. There are several common types of intervals:

1. Closed interval [a, b]:

   • Includes both endpoints $a$ and $b$.
   • Example: $[2, 5]$ represents all numbers from 2 to 5, including both 2 and 5.

2. Open interval (a, b):

   • Excludes both endpoints $a$ and $b$.
   • Example: $(2, 5)$ represents all numbers between 2 and 5, but not including 2 or 5.

3. Half-open (or half-closed) intervals:

   • One endpoint is included, and the other is excluded:
     • Left half-open interval [a, b): Includes $a$ but excludes $b$.
     • Right half-open interval (a, b]: Excludes $a$ but includes $b$.
   • Example: $[2, 5)$ includes 2 but not 5, while $(2, 5]$ includes 5 but not 2.

4. Infinite intervals:

   • These extend indefinitely in one or both directions:
     • Open infinite interval: $(-\infty, b)$ or $(a, \infty)$.
     • Closed infinite interval: $[a, \infty)$ or $(-\infty, b]$.

Let me know if you need more details or clarification on intervals!

Related Questions:

1. What are bounded and unbounded intervals?
2. How are intervals used in calculus (e.g., domain, range)?
3. How do you represent intervals in set notation?
4. What does an interval look like on a number line?
5. How do you solve inequalities involving intervals?

Tip:

When solving inequalities, intervals provide a great way to express the solution in a compact and clear manner!