1.1 Intervals
The set of real numbers, \(\mathbb {R}\) has 9 intervals. The first 4 kind’s are \([a,b]\hspace {0.2cm}, \hspace {0.2cm} (a,b)\hspace {0.2cm},\hspace {0.2cm} [a,b)\hspace {0.2cm},\hspace {0.2cm}(a,b]\)
The next 4 are described as follows: For each real number \(c\) there are four half-times given
as
\[ \{x \in \mathbb {R}:\, x \leq c\}\hspace {0.3cm},\hspace {0.3cm} \{x\in \mathbb {R}:\, x < c\}\]
\[\{x\in \mathbb {R}:\, x\geq c\}\hspace {0.3cm}, \hspace {0.3cm} \{x\in \mathbb {R}: \, x > c\}\]
The 9\(^{\text {th}}\) interval is the set \(\mathbb {R}\) itself.
Note.
- 1.
-
- \(\{x \in \mathbb {R}: \, x \leq c \} = (-\infty , c]\)
- \(\{x\in \mathbb {R}:\, x < c\} = (-\infty , c)\)
- \(\{x \in \mathbb {R}:\, x\geq c\} = [c,\infty )\)
- \(\{x\in \mathbb {R}: \, x > c\} = (c,\infty )\)
Where for every real number \(x\), \(\, -\infty < x < \infty \) and \(\infty \) is invented symbol, read as infinity \(\, \mathbb {R} = (-\infty , \infty )\).
- 2.
-
- (a)
- for \(a \in \mathbb {R}, \, (a,a) = \emptyset \hspace {0.2cm}\) Similarly, \([a,a) = \emptyset \, , \, (a,a]=\emptyset \)
- (b)
- \([a,a] = \{a\}\), a singleton set.
All these qualify to be intervals.
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