Certainly! Let's analyze the problem carefully and find the union and intersection of the two given sets E and F.
### Given Sets:
1. [tex]\( E = \{v \mid v \leq 1\} \)[/tex]
2. [tex]\( F = \{v \mid v < 8\} \)[/tex]
### Finding the Intersection [tex]\( E \cap F \)[/tex]:
The intersection of two sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex] includes all elements that are common to both sets.
1. Set [tex]\( E \)[/tex] includes all values less than or equal to 1.
2. Set [tex]\( F \)[/tex] includes all values less than 8.
Therefore, the values that satisfy both conditions (i.e., [tex]\( v \leq 1 \)[/tex] and [tex]\( v < 8 \)[/tex]) are the values in the interval from [tex]\(-\infty\)[/tex] to [tex]\(1\)[/tex], inclusive of 1.
So, the intersection [tex]\( E \cap F \)[/tex] is:
[tex]\[ E \cap F = (-\infty, 1] \][/tex]
### Finding the Union [tex]\( E \cup F \)[/tex]:
The union of two sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex] includes all elements that belong to either one of the sets or both.
1. Set [tex]\( E \)[/tex] includes all values less than or equal to 1.
2. Set [tex]\( F \)[/tex] includes all values less than 8.
The union includes all values that are less than 8 because:
- The values less than or equal to 1 (from set [tex]\( E \)[/tex]) are also included in the values less than 8 (from set [tex]\( F \)[/tex]).
So, the union [tex]\( E \cup F \)[/tex] is:
[tex]\[ E \cup F = (-\infty, 8) \][/tex]
### Summary:
The intersection and union of sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex] written in interval notation are:
- [tex]\( E \cap F = (-\infty, 1] \)[/tex]
- [tex]\( E \cup F = (-\infty, 8) \)[/tex]
