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Mathematical aspects of interval notation

The interval notation

In mathematics, an interval notation represents a pair of numbers. These numbers in an interval notation representation are the endpoints of an interval. Brackets or parenthesis can be used in an interval notation to show if these endpoints are included or excluded. For example (2, 8) is an interval of the real numbers that are within 2 and 8 where 2 is included but 8 is not included. An interval notation is simply another way of writing a range or a domain. In group builder notation braces (curly parentheses {}) and variables are used to show the range and domain. The interval notation is often considered more efficient. The symbols used in interval notation include:

  • Open parentheses ( )
  • Infinity ∞
  • Union Sign ∪
  • Closed parentheses [ ]
  • Negative Infinity −∞

In using the interval notation in a set of numbers: You will use the open parentheses ( ) only when the value is not included in the graph. This implies that the graph is undefined at that point or it has a hole or a jump or asymptote. If a graph goes beyond the left, the domain interval notation will begin with (−∞. If the graph goes downwards without stopping, the range interval notation will begin with (−∞. In a similar way, when the graph goes to the right or top without stopping, the interval notation will end with ∞). You will use the brackets [ ] in the interval notation only when the value is part of that graph. Any time a graph breaks, you should write the interval notation up to that point and then write another interval notation of the section of the graph that comes after that part and then inserts a union sign between every interval notation in order to merge them.


Algebraic inequalities like x≥4 implies that x is bigger than or equal to 4. Such an inequality has many solutions for x. Some of the solutions are 4, 5, 5.5, 6, 10 and so on. Since it is not possible to write down all the possible solutions for x, there are methods of expressing this infinite set of numbers. The most common methods used in mathematics to express these solutions of inequalities is graphing on a number line and using the interval notation.

To use an interval notation think of a group of numbers that are all larger than 5. If you are to write an inequality for such a group, you will start by letting x be any number in that group of numbers. It will be represented as x>5. This same group can be described in another form of notation which is the interval notation. In interval notation, the set of numbers will be expressed as: can be solved by the solutions of any of the two given inequalities. In interval notation, the solution set is a union of each of the solution. The compound inequalities that use the logical “and” are required to be solved by one solution. The solution set is an intersection of each solution. The compound inequalities of the form a<x

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