Step By Step Calculus » 2.3 - Sets of Numbers

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Synopsis
In the previous section we showed that the set of real numbers, namely \mathbb{R} \mathbb{R}, is a field. In addition, \mathbb{R} \mathbb{R} is ordered, that is we have the notion of one number being less than another number. In a geometric sense, the mathematical statement a < b a < b means that the number a a lies to the left of the number b b on the number line. For an algebraic definition, we need to intuitively understand which real numbers are positive and further understand that the statement a < b a < b means that \left(b-a\right) \left(b-a\right) is positive i.e. b-a > 0 b-a > 0. Note that the statement a < b a < b stands for strict inequality, which means it is not allowed that a=b a=b. If we want to include that possibility we write a \leq b a \leq b.
The real numbers form a totally ordered field, in that the statement a < b a < b is well-defined by the above, and in addition (“totally") for any two real numbers a a and b b, exactly one of the following is true: a < b, a = b a < b, a = b, or b < a b < a.
On the real line, we have the absolute value to measure distance. The absolute value of x x is written \mid x \mid \mid x \mid and is defined by
\displaystyle \mid x \mid = \left \{ \begin{array}{cc} x & \mbox{if} \; x \geq 0 , \\ -x & \mbox{if} \; x \leq 0 \end{array} \right.
\displaystyle \mid x \mid = \left \{ \begin{array}{cc} x & \mbox{if} \; x \geq 0 , \\ -x & \mbox{if} \; x \leq 0 \end{array} \right.
\mid x \mid \mid x \mid measures the distance of the number x x from the origin. Then \mid a - b \mid = \mid b - a \mid \mid a - b \mid = \mid b - a \mid measures the distance between a a and b b on the number line. For example, the distance between -2 -2 and 4 4 on the number line is \mid 4 - (-2) \mid = \mid -2 -4 \mid = 6 \; \mid 4 - (-2) \mid = \mid -2 -4 \mid = 6 \; .
In the previous section we used an intuitive definition of "sets". In this section we give a more complete treatment. We use the symbol \in \in ( reads as “in” or “belongs to") to specify set membership and \notin \notin (reads as “not in” or “ does not belong to") to specify non-membership. To mention all members of a given set A A we use the symbol \forall \forall (“for all”):      \forall x\in A \forall x\in A means “for all x x in the set A A” . To mention at least one member of a set A A we use the symbol \exists \exists (“there exists ”) :      \exists x\in A \exists x\in A means “there exists an x x in the set A A”). To indicate the negation of a symbol, we put a slash through it, for example \nexists \nexists means “there does not exist" :    \nexists \nexists icebergs in the Sahara desert.
Sets are denoted in a variety of ways, including, but not limited to:
  • Listing the elements in the set:   \left\{ 2,3,4, 5 \right\} \left\{ 2,3,4, 5 \right\} is the set consisting of the four natural numbers separated by commas inside the curly brackets.
  • Set-builder notation: Defining the set by a rule, in this case inequality constraints: \left\{ x \mid x \in \mathbb{N} : \right. \left\{ x \mid x \in \mathbb{N} : \right.\left.2 \leq x \leq 5 \right \} . \left.2 \leq x \leq 5 \right \} . Notice that this describes the same set as \left\{ 2,3,4, 5 \right\} \left\{ 2,3,4, 5 \right\} above.
  • Closed interval: Given two real numbers a a and b b with a<b a<b, the set of all real numbers lying between them, and including a a and b b, is written as [a,b] [a,b] using square brackets. This interval using set-builder notation is denoted by \{ x \mid x \in \mathbb{R}, \; a \leq x \leq b \} \{ x \mid x \in \mathbb{R}, \; a \leq x \leq b \}. Since this interval contains both endpoints it is called a closed interval.
  • Open interval:(a,b)(a,b), with parentheses is the notation for an open interval, which is an interval that does not contain either endpoint: \{ x \mid x \in \mathbb{R}, \; a < x < b \} \{ x \mid x \in \mathbb{R}, \; a < x < b \}.
  • Half-open/Half-closed interval:[a,b)[a,b) is also an interval, it includes a a (square bracket), but excludes b b (parenthesis). Similarly, (a,b] (a,b] includes b b but not a a. Both are called half-open or half-closed intervals.

    In the above figure, a solid dot means the corresponding point is in the set while an empty dot means the corresponding point is not in the set.
    An open interval is of one of four types : (-\infty,a), \; (a,b), \; (-\infty,a), \; (a,b), \; (b,\infty), (b,\infty), or (-\infty,\infty)(-\infty,\infty). An open interval has the important property that if x\in(a,b) x\in(a,b), then you can move any small distance from either direction of x x without leaving the interval.
    More generally, an open set is any set of real numbers which has the above property.