Definition:Real Interval/Notation/Unbounded Intervals

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Definition

Some authors (sensibly, perhaps) prefer not to use the $\infty$ symbol and instead use $\to$ and $\gets$ for $+\infty$ and $-\infty$ repectively.

In Wirth interval notation, such intervals are written as follows:

\(\displaystyle \hointr a \to\) \(:=\) \(\displaystyle \set {x \in \R: a \le x}\)
\(\displaystyle \hointl \gets a\) \(:=\) \(\displaystyle \set {x \in \R: x \le a}\)
\(\displaystyle \openint a \to\) \(:=\) \(\displaystyle \set {x \in \R: a < x}\)
\(\displaystyle \openint \gets a\) \(:=\) \(\displaystyle \set {x \in \R: x < a}\)
\(\displaystyle \openint \gets \to\) \(:=\) \(\displaystyle \set {x \in \R} = \R\)


In reverse-bracket notation, they appear as:

\(\displaystyle \left [{a, \to} \right]\) \(=\) \(\displaystyle \set {x \in \R: a \le x}\)
\(\displaystyle \left [{\gets, a} \right]\) \(=\) \(\displaystyle \set {x \in \R: x \le a}\)
\(\displaystyle \left ]{\, a, \to} \right]\) \(=\) \(\displaystyle \set {x \in \R: a < x}\)
\(\displaystyle \left [{\gets, a \,} \right[\) \(=\) \(\displaystyle \set {x \in \R: x < a}\)