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Subsets of real numbers

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.

 

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