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Sagot :
First, let's define the sets according to the given information:
1. Universal Set (U): [tex]\( U = \{ x \in \mathbb{N} : x \leq 8 \} = \{ 1, 2, 3, 4, 5, 6, 7, 8 \} \)[/tex]
2. Set A: [tex]\( A = \{ x \in \mathbb{N} : 5 < x^2 < 50 \} \)[/tex]
- We calculate for each element in [tex]\( U \)[/tex]:
- [tex]\( 1^2 = 1 \)[/tex] (not in range)
- [tex]\( 2^2 = 4 \)[/tex] (not in range)
- [tex]\( 3^2 = 9 \)[/tex] (in range)
- [tex]\( 4^2 = 16 \)[/tex] (in range)
- [tex]\( 5^2 = 25 \)[/tex] (in range)
- [tex]\( 6^2 = 36 \)[/tex] (in range)
- [tex]\( 7^2 = 49 \)[/tex] (in range)
- [tex]\( 8^2 = 64 \)[/tex] (not in range)
- So, [tex]\( A = \{3, 4, 5, 6, 7\} \)[/tex]
3. Set B: [tex]\( B = \{ x \in \mathbb{N} : x \text{ is a prime number less than 10} \} \)[/tex]
- The prime numbers less than 10 are 2, 3, 5, 7.
- So, [tex]\( B = \{2, 3, 5, 7\} \)[/tex]
Now, let's list the elements of the given sets.
i) [tex]\( A' \)[/tex]: The complement of A in [tex]\( U \)[/tex]:
- [tex]\( A' = U - A = \{ 1, 2, 8 \} \)[/tex]
ii) [tex]\( B' \)[/tex]: The complement of B in [tex]\( U \)[/tex]:
- [tex]\( B' = U - B = \{ 1, 4, 6, 8 \} \)[/tex]
iii) [tex]\( A - B \)[/tex]: Set difference of A and B:
- [tex]\( A - B = A \setminus B = \{ 4, 6 \} \)[/tex]
iv) [tex]\( A \cap B' \)[/tex]: Intersection of A and [tex]\( B' \)[/tex]:
- [tex]\( A \cap B' = A \cap (U - B) = \{ 4, 6 \} \)[/tex]
v) Is [tex]\( A - B = A \cap B' \)[/tex]?
- Yes, [tex]\( A - B = A \cap B' \)[/tex] because both are equal to \{ 4, 6 \}.
### Summary of the Sets:
- [tex]\( A' = \{ 1, 2, 8 \} \)[/tex]
- [tex]\( B' = \{ 1, 4, 6, 8 \} \)[/tex]
- [tex]\( A - B = \{ 4, 6 \} \)[/tex]
- [tex]\( A \cap B' = \{ 4, 6 \} \)[/tex]
- Yes, [tex]\( A - B = A \cap B' \)[/tex].
### Venn Diagram:
To visualize, you can draw a Venn diagram with a rectangle representing the universal set [tex]\( U \)[/tex], and two overlapping circles labeled [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. You would place the elements accordingly:
- [tex]\( A \)[/tex]: [tex]\( \{3, 4, 5, 6, 7\} \)[/tex]
- [tex]\( B \)[/tex]: [tex]\( \{2, 3, 5, 7\} \)[/tex]
- Overlap (intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]): [tex]\( \{3, 5, 7\} \)[/tex]
- Outside both sets (neither [tex]\( A \)[/tex] nor [tex]\( B \)[/tex]): [tex]\( \{1, 8\} \)[/tex]
You've now clearly demonstrated the relationships between the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], and listed the elements of the specific sets as requested.
1. Universal Set (U): [tex]\( U = \{ x \in \mathbb{N} : x \leq 8 \} = \{ 1, 2, 3, 4, 5, 6, 7, 8 \} \)[/tex]
2. Set A: [tex]\( A = \{ x \in \mathbb{N} : 5 < x^2 < 50 \} \)[/tex]
- We calculate for each element in [tex]\( U \)[/tex]:
- [tex]\( 1^2 = 1 \)[/tex] (not in range)
- [tex]\( 2^2 = 4 \)[/tex] (not in range)
- [tex]\( 3^2 = 9 \)[/tex] (in range)
- [tex]\( 4^2 = 16 \)[/tex] (in range)
- [tex]\( 5^2 = 25 \)[/tex] (in range)
- [tex]\( 6^2 = 36 \)[/tex] (in range)
- [tex]\( 7^2 = 49 \)[/tex] (in range)
- [tex]\( 8^2 = 64 \)[/tex] (not in range)
- So, [tex]\( A = \{3, 4, 5, 6, 7\} \)[/tex]
3. Set B: [tex]\( B = \{ x \in \mathbb{N} : x \text{ is a prime number less than 10} \} \)[/tex]
- The prime numbers less than 10 are 2, 3, 5, 7.
- So, [tex]\( B = \{2, 3, 5, 7\} \)[/tex]
Now, let's list the elements of the given sets.
i) [tex]\( A' \)[/tex]: The complement of A in [tex]\( U \)[/tex]:
- [tex]\( A' = U - A = \{ 1, 2, 8 \} \)[/tex]
ii) [tex]\( B' \)[/tex]: The complement of B in [tex]\( U \)[/tex]:
- [tex]\( B' = U - B = \{ 1, 4, 6, 8 \} \)[/tex]
iii) [tex]\( A - B \)[/tex]: Set difference of A and B:
- [tex]\( A - B = A \setminus B = \{ 4, 6 \} \)[/tex]
iv) [tex]\( A \cap B' \)[/tex]: Intersection of A and [tex]\( B' \)[/tex]:
- [tex]\( A \cap B' = A \cap (U - B) = \{ 4, 6 \} \)[/tex]
v) Is [tex]\( A - B = A \cap B' \)[/tex]?
- Yes, [tex]\( A - B = A \cap B' \)[/tex] because both are equal to \{ 4, 6 \}.
### Summary of the Sets:
- [tex]\( A' = \{ 1, 2, 8 \} \)[/tex]
- [tex]\( B' = \{ 1, 4, 6, 8 \} \)[/tex]
- [tex]\( A - B = \{ 4, 6 \} \)[/tex]
- [tex]\( A \cap B' = \{ 4, 6 \} \)[/tex]
- Yes, [tex]\( A - B = A \cap B' \)[/tex].
### Venn Diagram:
To visualize, you can draw a Venn diagram with a rectangle representing the universal set [tex]\( U \)[/tex], and two overlapping circles labeled [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. You would place the elements accordingly:
- [tex]\( A \)[/tex]: [tex]\( \{3, 4, 5, 6, 7\} \)[/tex]
- [tex]\( B \)[/tex]: [tex]\( \{2, 3, 5, 7\} \)[/tex]
- Overlap (intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]): [tex]\( \{3, 5, 7\} \)[/tex]
- Outside both sets (neither [tex]\( A \)[/tex] nor [tex]\( B \)[/tex]): [tex]\( \{1, 8\} \)[/tex]
You've now clearly demonstrated the relationships between the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], and listed the elements of the specific sets as requested.
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