Intervals as subset | Part-1

Intervals as Subsets of Real Numbers

In mathematics, an interval refers to a specific kind of subset within the set of real numbers (( mathbb{R} )), made up of all values located between two given endpoints. Intervals are fundamental in calculus, analysis, and many areas of mathematics because they represent continuous sets of numbers without gaps.

Types of Intervals

Intervals are categorized based on whether they include or exclude their endpoints. The main types of intervals are:

• Closed Interval ([a, b]): Includes both endpoints (a) and (b). It includes every real number x where x falls between a and b, including the endpoints.

Example: ([2, 5] = { x in mathbb{R} mid 2 leq x leq 5 })

• Open Interval ((a, b)): Excludes both endpoints (a) and (b). It contains all real numbers (x) such that (a < x < b).

Example: ((0, 1) = { x in mathbb{R} mid 0 < x < 1 })

• Half-Open (or Half-Closed) Intervals:
• ([a, b)): Includes (a) but excludes (b), where (a leq x < b).

Example: ([3, 7) = { x in mathbb{R} mid 3 leq x < 7 })

• ((a, b]): Excludes (a) but includes (b), where (a < x leq b).

Example: ((1, 4] = { x in mathbb{R} mid 1 < x leq 4 })

• Infinite Intervals: Intervals can extend indefinitely in one or both directions.
• ([a, infty) = { x in mathbb{R} mid x geq a })
• ((-infty, b) = { x in mathbb{R} mid x < b })
• ((-infty, infty) = mathbb{R}), the entire set of real numbers.

Intervals as Subsets of Real Numbers

Every interval forms a continuous subset of real numbers, with no interruptions or gaps between the values it contains. This continuity distinguishes intervals from arbitrary subsets that might consist of disconnected points.

For example, the interval ([1, 3]) includes all numbers between 1 and 3, such as 1.5, 2, and 2.999, without any gaps. On the other hand, a set like ({1, 2, 3}) contains only specific points and is not an interval.

Notation Summary

Applications

Intervals are widely used in mathematics, physics, engineering, and economics for defining domains of functions, integration limits, and describing ranges of variables. Their continuous nature makes them crucial for calculus and real analysis.

Intervals serve as a key concept for representing continuous subsets of real numbers. By understanding the different types of intervals and their notation, one can better grasp various mathematical concepts and their applications.