Intervals.

$\displaystyle (a,b)$ $\displaystyle =\{ x \in \mathbb{R} \; \vert \; a<x<b \} \qquad ($open interval)$\displaystyle .$    
$\displaystyle [a,b]$ $\displaystyle =\{ x \in \mathbb{R} \; \vert \; a \leq x \leq b \} \qquad ($closed interval$\displaystyle ).$    
$\displaystyle (a,b]$ $\displaystyle =\{ x \in \mathbb{R} \; \vert \; a < x \leq b \}.$    
$\displaystyle [a,b)$ $\displaystyle =\{ x \in \mathbb{R} \; \vert \; a \leq x<b \}.$    
$\displaystyle (-\infty , b]$ $\displaystyle = \{ x \in \mathbb{R} \; \vert \; x \leq b \}.$    
$\displaystyle (-\infty , b)$ $\displaystyle = \{ x \in \mathbb{R} \; \vert \; x < b \}.$    
$\displaystyle [a, + \infty)$ $\displaystyle = \{ x \in \mathbb{R} \; \vert \; a \leq x \}.$    
$\displaystyle (a, + \infty)$ $\displaystyle = \{ x \in \mathbb{R} \; \vert \; a<x \}.$    
$\displaystyle (-\infty, + \infty )$ $\displaystyle = \mathbb{R}.$    

Example 1.3.1  

$\displaystyle \mathbb{R}- \{-1,3 \}$ $\displaystyle =(- \infty, -1 ) \cup (-1, 3) \cup (3, + \infty )$    
$\displaystyle (2,6)$ $\displaystyle = \{ x \in \mathbb{R} \; ; \; 2 < x < 6 \}$   cf Fig. 1(a)    
$\displaystyle \mathbb{R}^*$ $\displaystyle =(- \infty , 0) \cup (0,+ \infty )$    
$\displaystyle [-1,+ \infty )$ $\displaystyle = \{ x \in \mathbb{R} \; ; \; x \geq -1 \}$   cf Fig. 1(b)    
$\displaystyle \mathbb{R}^{+}$ $\displaystyle = [0, + \infty )$    
$\displaystyle \mathbb{R}^{-}$ $\displaystyle = (- \infty , 0]$    
$\displaystyle \mathbb{R}^{+*}$ $\displaystyle = (0, + \infty )$    
$\displaystyle \mathbb{R}^{-*}$ $\displaystyle = (- \infty , 0)$    

Figure 1: Intervals.
\begin{figure}\mbox{
\subfigure[$(2,6) $]{\epsfig{file=OpenInterval.eps,width=6...
...[$[-1,+ \infty )$]{\epsfig{file=InfiniteInterv.eps,width=6cm}
}
}\end{figure}

Noah Dana-Picard 2007-12-28