To determine the value of [tex]\( n(A \cap B) \)[/tex], we need to find the number of elements in the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Step-by-step, we follow these steps:
1. Identify the elements in set [tex]\( A \)[/tex] and set [tex]\( B \)[/tex]:
[tex]\[
A = \{2, 3, 5\}
\][/tex]
[tex]\[
B = \{1, 4\}
\][/tex]
2. Find the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]: The intersection [tex]\( A \cap B \)[/tex] contains all elements that are both in [tex]\( A \)[/tex] and in [tex]\( B \)[/tex].
In this case, we check each element of [tex]\( A \)[/tex] to see if it is also in [tex]\( B \)[/tex]:
- [tex]\( 2 \)[/tex] is in [tex]\( A \)[/tex] but not in [tex]\( B \)[/tex].
- [tex]\( 3 \)[/tex] is in [tex]\( A \)[/tex] but not in [tex]\( B \)[/tex].
- [tex]\( 5 \)[/tex] is in [tex]\( A \)[/tex] but not in [tex]\( B \)[/tex].
Similarly, we can check the elements of [tex]\( B \)[/tex]:
- [tex]\( 1 \)[/tex] is in [tex]\( B \)[/tex] but not in [tex]\( A \)[/tex].
- [tex]\( 4 \)[/tex] is in [tex]\( B \)[/tex] but not in [tex]\( A \)[/tex].
None of the elements from set [tex]\( A \)[/tex] are present in set [tex]\( B \)[/tex]. Therefore, the intersection [tex]\( A \cap B \)[/tex] is the empty set:
[tex]\[
A \cap B = \emptyset
\][/tex]
3. Determine [tex]\( n(A \cap B) \)[/tex]: [tex]\( n(A \cap B) \)[/tex] represents the number of elements in the intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. Since the intersection [tex]\( A \cap B \)[/tex] is an empty set, it contains no elements, so:
[tex]\[
n(A \cap B) = 0
\][/tex]
Hence, the value of [tex]\( n(A \cap B) \)[/tex] is [tex]\( 0 \)[/tex].
Therefore, the answer is:
[tex]\[
\boxed{0}
\][/tex]
So, the correct option is:
[tex]\( (c) 0 \)[/tex]
