A={1,3,6,9},B={4,5,6,9},C={x:5<x<_9} are subset of the universal set U = {1,2,3,4,5,6,7,8,9} Find:
(I) A n (B n C)
(II)(A U B)n(B U C)
well, you know that 5 < x ≤ 9 so x is an element of {6,7,8,9}
(A U B) n (B U C) = {1, 3, 4, 5, 6, 9} n {4, 5, 6, 9, 6, 7, 8, 9} = {4,5,6,9}
Yes, you are correct. I apologize for overlooking that information. Given that 5 < x ≤ 9 and x is an element of {6,7,8,9}, the intersection of (A U B) and (B U C) is indeed {4, 5, 6, 9}. Thank you for providing that clarification.
(I) To find A n (B n C), we need to first find B n C, and then find the intersection of A with the result.
1. B n C: We need to find the elements that are common to both sets B and C.
Set B = {4, 5, 6, 9} and Set C = {x: 5 < x ≤ 9}.
Comparing the two sets, we can see that the common element between Set B and Set C is 6, as it is the only element that satisfies both conditions.
Therefore, B n C = {6}.
2. A n (B n C): Now, we need to find the elements that are common to sets A and B n C.
Set A = {1, 3, 6, 9} and Set B n C = {6}.
Comparing the two sets, we can see that the common element between Set A and Set B n C is also 6.
Therefore, A n (B n C) = {6}.
Final answer: A n (B n C) = {6}.
(II) To find (A U B) n (B U C), we first need to find the union of sets A and B, and then find the intersection of the result with the union of sets B and C.
1. A U B: We need to find all the elements that are present in sets A and B combined.
Set A = {1, 3, 6, 9} and Set B = {4, 5, 6, 9}.
The union of A and B is {1, 3, 4, 5, 6, 9}.
2. B U C: We need to find all the elements that are present in sets B and C combined.
Set B = {4, 5, 6, 9} and Set C = {x: 5 < x ≤ 9}.
The union of B and C will be the same as Set C, as all the elements in Set B are also in Set C.
Therefore, B U C = {x: 5 < x ≤ 9}.
3. (A U B) n (B U C): Now, we need to find the elements that are common to the sets (A U B) and (B U C).
Set (A U B) = {1, 3, 4, 5, 6, 9} and Set (B U C) = {x: 5 < x ≤ 9}.
Comparing the two sets, we can see that the common elements are 5, 6, and 9.
Therefore, (A U B) n (B U C) = {5, 6, 9}.
Final answer: (A U B) n (B U C) = {5, 6, 9}.
(I) A n (B n C)
To find the intersection of sets, we need to find elements that are common to all sets.
B n C = {6, 9} (since 6 and 9 are the only elements in both set B and set C)
A n (B n C) = {1, 3, 6, 9} n {6, 9} = {6, 9}
Therefore, A n (B n C) = {6, 9}.
(II) (A U B) n (B U C)
To find the union of sets, we combine all elements from both sets.
A U B = {1, 3, 4, 5, 6, 9} (as it contains all elements from both sets A and B)
B U C = {4, 5, 6, 9, x: 5 < x ≤ 9} (as it contains all elements from set B and any element between 5 and 9)
(A U B) n (B U C) = {1, 3, 4, 5, 6, 9} n {4, 5, 6, 9, x: 5 < x ≤ 9}
Simplifying this expression requires additional information about the specific value of "x" within the given condition. Without that information, the intersection cannot be determined precisely.