\( F \) and \( H \) are sets of real numbers defined as follows. \[ \begin{array}{l} F=\{y \mid y \leq 3\} \\ H=\{y \mid y>7\} \end{array} \] Write \( F \cup H \) and \( F \cap H \) using interval notation. If the set is empty, write \( \varnothing \). \[ F \cup H= \] \[ F \cap H= \] \( \square \) \( \infty \) \( -\infty \)

To find the union and intersection of the sets \( F \) and \( H \), we analyze their definitions. The set \( F = \{y \mid y \leq 3\} \) represents all real numbers less than or equal to 3, which can be written in interval notation as \( (-\infty, 3] \). The set \( H = \{y \mid y > 7\} \) includes all real numbers greater than 7, represented as \( (7, \infty) \). Now, for the union \( F \cup H \), we combine both intervals: \[ F \cup H = (-\infty, 3] \cup (7, \infty) \] For the intersection \( F \cap H \), we look for values that are in both sets simultaneously. Since \( F \) contains numbers that are less than or equal to 3 and \( H \) contains numbers greater than 7, there are no overlaps, resulting in an empty set: \[ F \cap H = \varnothing \] Thus, the final answers are: \[ F \cup H = (-\infty, 3] \cup (7, \infty) \] \[ F \cap H = \varnothing \]