If U=( 1,2,3,4,5,6,7,8,9,10) A=(1,2,4,5) B=(2,4,6,8) C=(2,3,5,7,9) find i) AuB ii) AnBnC iii) BnC iv)A' v)B' vi) A'nB' vii) A'uC'
i) AuB is the union of sets A and B. To find this, we combine all the elements from both sets, removing any duplicates:
U = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
A = (1, 2, 4, 5)
B = (2, 4, 6, 8)
Combining A and B, we get:
AuB = (1, 2, 4, 5, 6, 8)
ii) AnBnC is the intersection of sets A, B, and C. To find this, we identify the elements that are common to all three sets:
C = (2, 3, 5, 7, 9)
AnBnC = (2)
iii) BnC is the intersection of sets B and C. To find this, we identify the elements that are common to both sets:
BnC = (2)
iv) A' is the complement of set A, which means it includes all the elements in U that are not in A. To find this, we subtract the elements of A from U:
A' = (3, 6, 7, 8, 9, 10)
v) B' is the complement of set B, which means it includes all the elements in U that are not in B. To find this, we subtract the elements of B from U:
B' = (1, 3, 5, 7, 9, 10)
vi) A'nB' is the intersection of the complement of A and the complement of B. To find this, we identify the elements that are common to both sets:
A'nB' = (3, 7, 9, 10)
vii) A'uC' is the union of the complement of A and the complement of C. To find this, we combine all the elements from both sets, removing any duplicates:
A' = (3, 6, 7, 8, 9, 10)
C' = (1, 4, 6, 8, 10)
Combining A' and C', we get:
A'uC' = (1, 3, 4, 6, 7, 8, 9, 10)
To find the solutions for each question, we will go step-by-step:
i) To find AuB (the union of A and B), we need to combine all elements from both sets without any repetitions.
Therefore,
AuB = {1, 2, 4, 5, 6, 8}
ii) To find AnBnC (the intersection of A, B, and C), we need to find the common elements in all three sets.
AnBnC = {}
iii) To find BnC (the intersection of B and C), we need to find the common elements in both sets.
BnC = {2}
iv) To find A' (the complement of A), we need to find all elements in the universal set U that are not in A.
A' = {3, 6, 7, 8, 9, 10}
v) To find B' (the complement of B), we need to find all elements in the universal set U that are not in B.
B' = {1, 3, 5, 7, 9, 10}
vi) To find A'nB' (the intersection of A' and B'), we need to find the common elements in both sets.
A'nB' = {3, 5, 7, 9, 10}
vii) To find A'uC' (the union of A' and C'), we need to combine all elements from both sets without any repetitions.
A'uC' = {1, 3, 5, 7, 9, 10}
To compute the given set operations, we need to understand the basic principles of set theory. Here's how you can find the answers step by step:
Given:
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 2, 4, 5}
B = {2, 4, 6, 8}
C = {2, 3, 5, 7, 9}
i) A ∪ B (Union of A and B)
To find the union of sets A and B, we combine all the elements from both sets while eliminating any duplicates:
A ∪ B = {1, 2, 4, 5, 6, 8}
ii) A ∩ B ∩ C (Intersection of A, B, and C)
For the intersection of A, B, and C, we find the common elements present in all three sets:
A ∩ B ∩ C = {2}
iii) B ∩ C (Intersection of B and C)
To find the intersection of sets B and C, we look for the common elements between the two sets:
B ∩ C = {2}
iv) A' (Complement of A)
To find the complement of set A, we need to identify the elements in U that are not present in A:
A' = {3, 6, 7, 8, 9, 10}
v) B' (Complement of B)
To find the complement of set B, we identify the elements in U that are not present in B:
B' = {1, 3, 5, 7, 9, 10}
vi) A' ∩ B' (Intersection of A' and B')
To find the intersection of sets A' and B', we look for the common elements between the two sets:
A' ∩ B' = {3, 7, 9, 10}
vii) A' ∪ C' (Union of A' and C')
To find the union of sets A' and C', we combine all the elements from both sets while eliminating any duplicates:
A' ∪ C' = {1, 3, 6, 7, 9, 10}
Remember to always verify your answer by double-checking the calculations.