Let's analyze the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] and determine their intersection.
1. Define the Sets:
- Set [tex]\( A \)[/tex] contains all [tex]\( x \)[/tex] such that [tex]\( x < 1 \)[/tex].
- Set [tex]\( B \)[/tex] contains all [tex]\( x \)[/tex] such that [tex]\( x > 3 \)[/tex].
2. Describe the Elements in Each Set:
- For [tex]\( A \)[/tex], the elements are any numbers that are less than 1.
- For [tex]\( B \)[/tex], the elements are any numbers that are greater than 3.
3. Intersection of Sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- The intersection [tex]\( A \cap B \)[/tex] consists of all elements that are in both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
- An element [tex]\( x \)[/tex] has to satisfy both conditions: [tex]\( x < 1 \)[/tex] and [tex]\( x > 3 \)[/tex].
4. Analyze the Conditions:
- A number [tex]\( x \)[/tex] cannot be simultaneously less than 1 and greater than 3.
- There are no numbers that can fulfill both conditions at the same time.
Therefore, there are no elements that belong to both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], meaning the intersection [tex]\( A \cap B \)[/tex] is the empty set.
Hence, the answer is [tex]\(\varnothing\)[/tex].
