Let's analyze the sets [tex]\( F \)[/tex] and [tex]\( H \)[/tex]:
- The set [tex]\( F \)[/tex] is defined as [tex]\( F = \{ y \mid y \leq 1 \} \)[/tex]. In interval notation, this is written as [tex]\( (-\infty, 1] \)[/tex], which includes all real numbers less than or equal to 1.
- The set [tex]\( H \)[/tex] is defined as [tex]\( H = \{ y \mid y > 8 \} \)[/tex]. In interval notation, this is written as [tex]\( (8, \infty) \)[/tex], which includes all real numbers greater than 8.
To find the intersection [tex]\( F \cap H \)[/tex]:
- The intersection of two sets is the set of elements that are common to both sets.
- Set [tex]\( F \)[/tex] consists of numbers less than or equal to 1, while set [tex]\( H \)[/tex] consists of numbers greater than 8.
- Since there are no numbers that can simultaneously be less than or equal to 1 and greater than 8, the intersection is empty.
Thus, [tex]\( F \cap H = \varnothing \)[/tex].
Next, to find the union [tex]\( F \cup H \)[/tex]:
- The union of two sets is the set of all elements that are in either set.
- Set [tex]\( F \)[/tex] includes all numbers less than or equal to 1, i.e., [tex]\( (-\infty, 1] \)[/tex].
- Set [tex]\( H \)[/tex] includes all numbers greater than 8, i.e., [tex]\( (8, \infty) \)[/tex].
- The union of these two sets combines all numbers in [tex]\( (-\infty, 1] \)[/tex] with all numbers in [tex]\( (8, \infty) \)[/tex].
Thus, [tex]\( F \cup H = (-\infty, 1] \cup (8, \infty) \)[/tex].
In conclusion:
[tex]\[
F \cap H = \varnothing
\][/tex]
[tex]\[
F \cup H = (-\infty, 1] \cup (8, \infty)
\][/tex]
