To solve this problem, we'll go through each set and identify where the integers 1 to 15 belong in the Venn diagram. We have two sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] and their universal set is composed of the numbers from 1 to 15.
Given Sets:
- [tex]\( A = \{1, 2, 3, 4, 5, 6, 7\} \)[/tex]
- [tex]\( B = \{5, 6, 7, 8, 9, 10, 11\} \)[/tex]
We need to place the integers into four categories based on these sets:
1. Intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]: Elements in both [tex]\( A \)[/tex] and [tex]\( B \)[/tex]
2. Only in [tex]\( A \)[/tex]: Elements in [tex]\( A \)[/tex] but not in [tex]\( B \)[/tex]
3. Only in [tex]\( B \)[/tex]: Elements in [tex]\( B \)[/tex] but not in [tex]\( A \)[/tex]
4. Complement: Elements that are neither in [tex]\( A \)[/tex] nor in [tex]\( B \)[/tex]
Step-by-Step Breakdown:
1. Intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
Elements that are in both sets:
- Intersection: [tex]\( \{5, 6, 7\} \)[/tex]
2. Only in [tex]\( A \)[/tex]:
Elements that are in [tex]\( A \)[/tex] but not in [tex]\( B \)[/tex]:
- Only in [tex]\( A \)[/tex]: [tex]\( \{1, 2, 3, 4\} \)[/tex]
3. Only in [tex]\( B \)[/tex]:
Elements that are in [tex]\( B \)[/tex] but not in [tex]\( A \)[/tex]:
- Only in [tex]\( B \)[/tex]: [tex]\( \{8, 9, 10, 11\} \)[/tex]
4. Complement of the union of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
Elements that are in the universal set but not in the union of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- Universal set: [tex]\( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\} \)[/tex]
- Union of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]: [tex]\( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\} \)[/tex]
- Complement: [tex]\( \{12, 13, 14, 15\} \)[/tex]
Final Placement in the Venn Diagram:
- Intersection (A ∩ B): [tex]\( \{5, 6, 7\} \)[/tex]
- Numbers: 5, 6, 7
- Only in [tex]\( A \)[/tex]: [tex]\( \{1, 2, 3, 4\} \)[/tex]
- Numbers: 1, 2, 3, 4
- Only in [tex]\( B \)[/tex]: [tex]\( \{8, 9, 10, 11\} \)[/tex]
- Numbers: 8, 9, 10, 11
- Complement (neither in [tex]\( A \)[/tex] nor [tex]\( B \)[/tex]): [tex]\( \{12, 13, 14, 15\} \)[/tex]
- Numbers: 12, 13, 14, 15
Thus, the numbers in the Venn diagram should be arranged as follows:
- Inside the intersection (middle part where [tex]\( A \)[/tex] and [tex]\( B \)[/tex] overlap): 5, 6, 7.
- In set [tex]\( A \)[/tex] only: 1, 2, 3, 4.
- In set [tex]\( B \)[/tex] only: 8, 9, 10, 11.
- Outside the Venn diagram (complement): 12, 13, 14, 15.
This placement groups the numbers 1 through 15 according to their membership in sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
