Subsets of real numbers especially intervals (with notation)
Many quantities in this world that is real can be quantified with the help of real numbers. These could include temperature at a specific time period, revenue that may be generated by the sale of certain products or even the maximum population of Sasquatch that could inhabit a specific region.
Subsets of Real Numbers
Listed below are few special types of subsets of real numbers that are worthy of a special note.
- Natural Numbers: N={1, 2, 3, …} Here, the ellipse ‘…’ indicates that natural numbers include 1, 2, 3 and forth one that would follow.
- Whole Numbers: W= {0, 1, 2, …}.
- Integers: V = {… , 3, 2, 1, 0, 1, 2, 3, …} = {0, ±1, ±2, ±3, …}. a
- Rational Numbers: Q = a b | a 2 Z and b 2 Z. Rational Numbers are said to be the ratios of integers for whom the denominator is not a zero. With this, another appropriate way to represent rational numbers would be: Q = {x | x possesses a terminating or repeating decimal representation}.
- Irrational Numbers: P = {x | x 2 R but x 2/ Q}c. is a actually a real number which isn’t a rational number. When said in a different manner, P = {x | x possesses a decimal representation which is neither repeating nor terminating}.
Interval Notation
Consider a, b εR and a < b. Now the entire set of real numbers that fall between a and b would be denoted in the interval notation form like (a,b) and would be determined as an open interval.
Thus in that case, (a,b) = { x εR : a < x < b).
This (a,b) would be recognized as an interval of all the real numbers that are included between a and b and exclude both a and b. All points between a and b would belong to the open interval i.e. (a,b) however, a and b are not supposed to belong to it.
The intervals that contain the end points are called closed intervals and denoted as [a,b] replacing brackets by parenthesis.
[a,b] = { x εR : a ≤ x ≤ b }
Here, [a.b] is the interval of all real numbers that are included between a and b; and include a and b both. There could also be intervals that are closed at one end and left open on the other end.
[a,b) = { x εR : a ≤ x < b}
Here, [a,b) is the interval of real numbers falling between a and b and include a however, exclude b.
(a,b] = { x εR : a < x ≤ b}
Similarly, in the above example, (a,b] is the interval of real numbers falling between a and b. They exclude a and include b.
In interval notations therefore, a set of positive real numbers R+ could be written as R+ = (0,∞). In the same, a set of negative real numbers is written as (-∞ ,0). The set R in itself as an interval notation is known to be given as (-∞ ,∞). Also, the length of any interval [a,b) or (a,b] or [a,b] or interval (a,b) is known to be given as b – a.
