$F$ and $H$ are sets of real numbers defined as follows. \[ \begin{array}{l} F=\{v \mid v<1\} \\ H=\{v \mid v \leq 7\} \end{array} \] Write $F \cup H$ and $F \cap H$ using interval notation. If the set is empty, write 0. \[ \begin{array}{l} f \cup n=\square \\ F \cap n=\square \end{array} \]
Solution
Step 1 ：Define the sets $F$ and $H$ as $F=\{v \mid v<1\}$ and $H=\{v \mid v \leq 7\}$ respectively.
Step 2 ：The union of two sets is the set of elements that are in either set. So, for the union $F \cup H$, since $F$ is the set of all real numbers less than 1 and $H$ is the set of all real numbers less than or equal to 7, the union would be the set of all real numbers less than or equal to 7. This is because 7 is the larger of the two upper bounds.
Step 3 ：The intersection of two sets is the set of elements that are in both sets. So, for the intersection $F \cap H$, since $F$ is the set of all real numbers less than 1 and $H$ is the set of all real numbers less than or equal to 7, the intersection would be the set of all real numbers less than 1. This is because 1 is the smaller of the two upper bounds.
Step 4 ：Write the final answer in interval notation. So, $F \cup H=(-\infty, 7]$ and $F \cap H=(-\infty, 1)$
Step 5 ：\(\boxed{F \cup H=(-\infty, 7], F \cap H=(-\infty, 1)}\)
