Certainly! Let's walk through the problem step-by-step.
Given sets:
- [tex]\( E = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)[/tex]
- [tex]\( A = \{2, 3, 5, 7\} \)[/tex]
- [tex]\( B = \{4, 6, 8, 10\} \)[/tex]
We are asked to explain why [tex]\( A \cap B = \varnothing \)[/tex].
### Step-by-step solution:
1. Understand the concept of set intersection ([tex]\( \cap \)[/tex]):
- The intersection of two sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] (denoted [tex]\( A \cap B \)[/tex]) is the set of all elements that are both in [tex]\( A \)[/tex] and in [tex]\( B \)[/tex].
2. Identify elements of [tex]\( A \)[/tex]:
- [tex]\( A = \{2, 3, 5, 7\} \)[/tex]
3. Identify elements of [tex]\( B \)[/tex]:
- [tex]\( B = \{4, 6, 8, 10\} \)[/tex]
4. Find common elements between [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- Compare each element of set [tex]\( A \)[/tex] with each element of set [tex]\( B \)[/tex]:
- [tex]\( 2 \)[/tex] (in [tex]\( A \)[/tex]) is not in [tex]\( B \)[/tex]
- [tex]\( 3 \)[/tex] (in [tex]\( A \)[/tex]) is not in [tex]\( B \)[/tex]
- [tex]\( 5 \)[/tex] (in [tex]\( A \)[/tex]) is not in [tex]\( B \)[/tex]
- [tex]\( 7 \)[/tex] (in [tex]\( A \)[/tex]) is not in [tex]\( B \)[/tex]
- [tex]\( 4 \)[/tex] (in [tex]\( B \)[/tex]) is not in [tex]\( A \)[/tex]
- [tex]\( 6 \)[/tex] (in [tex]\( B \)[/tex]) is not in [tex]\( A \)[/tex]
- [tex]\( 8 \)[/tex] (in [tex]\( B \)[/tex]) is not in [tex]\( A \)[/tex]
- [tex]\( 10 \)[/tex] (in [tex]\( B \)[/tex]) is not in [tex]\( A \)[/tex]
5. Conclusion:
- There are no elements that are common to both set [tex]\( A \)[/tex] and set [tex]\( B \)[/tex]. This means that the intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is an empty set.
Therefore, we write:
[tex]\[ A \cap B = \varnothing \][/tex]
This means that there are no elements that belong to both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], and thus, [tex]\( A \cap B \)[/tex] is an empty set.
