To solve the problem, let's analyze each statement given the condition [tex]\( A \cap B = \varnothing \)[/tex]. This means that sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] have no elements in common.
1. Statement: [tex]\( A \)[/tex] is equal to [tex]\( B \)[/tex] complement.
- The complement of [tex]\( B \)[/tex] ([tex]\( B^c \)[/tex]) consists of all elements that are not in [tex]\( B \)[/tex].
- Since [tex]\( A \cap B = \varnothing \)[/tex], all elements of [tex]\( A \)[/tex] are not in [tex]\( B \)[/tex]. Therefore, [tex]\( A \subseteq B^c \)[/tex].
- Given that the universal set contains only [tex]\( A \)[/tex] and [tex]\( B \)[/tex], and assuming we are dealing with the entire universal set, we would typically view [tex]\( A \)[/tex] as [tex]\( U - B \)[/tex].
- So, it is reasonable to consider that [tex]\( A = B^c \)[/tex].
2. Statement: [tex]\( A \cup B = 0 \)[/tex].
- The union of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] ([tex]\( A \cup B \)[/tex]) includes all elements that are in either [tex]\( A \)[/tex] or [tex]\( B \)[/tex].
- Since [tex]\( A \cap B = \varnothing \)[/tex], [tex]\( A \cup B \)[/tex] cannot be empty unless both [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are empty, which contradicts the premise unless explicitly stated.
- This statement is false because [tex]\( A \cup B \)[/tex] should include the combined elements from both [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
3. Statement: [tex]\( A \)[/tex] is not a subset of [tex]\( B \)[/tex].
- A subset means that every element of [tex]\( A \)[/tex] is also an element of [tex]\( B \)[/tex] ([tex]\( A \subseteq B \)[/tex]).
- Given [tex]\( A \cap B = \varnothing \)[/tex], no elements of [tex]\( A \)[/tex] are in [tex]\( B \)[/tex]. This directly implies that [tex]\( A \)[/tex] cannot be a subset of [tex]\( B \)[/tex].
- Thus, [tex]\( A \not\subseteq B \)[/tex] is a true statement.
4. Statement: [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are disjoint.
- Disjoint sets are sets that have no elements in common.
- By definition, [tex]\( A \cap B = \varnothing \)[/tex] directly tells us that [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are disjoint sets.
- This statement is true.
By evaluating all the statements thoroughly, we can conclude that the only statement that is false is:
- Statement 2: [tex]\( A \cup B = 0 \)[/tex]
So, the correct answer is:
2
