Write the set $\{x \mid x
eq 2\}$ using interval notation.
The set notation \(\{x \mid x eq 2\}\) represents the collection of all real numbers \(x\) except for the number 2. On the real number line, this set consists of all values to the left of 2 and all values to the right of 2. In interval notation, we represent "all values less than 2" as the open interval \((-\infty, 2)\), and "all values greater than 2" as the open interval \((2, +\infty)\). To combine these two separate intervals into one set, we use the union symbol \(\cup\). Following the specified formatting requirements, we use "U" for the union symbol, "-oo" for negative infinity, and "+oo" for positive infinity. Therefore, the set is written as \((-\infty, 2) U (2, +\infty)\).
Interval notation is a standard way to define domains of functions or ranges of values in science and engineering. For instance, if a machine can operate safely at any temperature except for a specific critical temperature of \(200^\circ C\), we could describe the safe operating range as \(T \in (-\infty, 200) U (200, +\infty)\). Learning this notation helps in precisely defining conditions for physics problems, economic models, and statistical analysis.