Giving that U={1,2,3,4,5,6,7,8,9,10} p={1,2,4,6,10} Q={2,3,6,9}
Find
(a). PUQ
(b). PNQ
(C).p'UQ'
(d).n[pUQ]
(e).p'nQ'
(a). PUQ refers to the union of sets P and Q. The union of sets is the combination of all distinct elements from both sets.
PUQ = {1, 2, 3, 4, 6, 9, 10}
(b). PNQ refers to the intersection of sets P and Q. The intersection of sets refers to the common elements between both sets.
PNQ = {2, 6}
(C). p'UQ' refers to the complement of set P union with the complement of set Q. The complement of a set contains all the elements that are not in the original set.
p'UQ' = {3, 5, 7, 8, 9}
(d). n[pUQ] refers to the set difference between set P union with set Q and the universal set U. It includes all the elements in P and Q that are not in the universal set U.
n[pUQ] = {}
(e). p'nQ' refers to the set difference between set P and set Q union with their complements. It includes all the elements in set P that are not in set Q, and are also not in the complement of set Q.
p'nQ' = {1, 4, 10}
To find the given operations, we will need to understand the basic principles of set theory. Let's go step-by-step:
(a) To find PUQ:
PUQ represents the union of sets P and Q, which includes all elements that are in either set P or set Q, or in both.
PUQ = {1, 2, 4, 6, 9, 10}
Explanation:
- From set P, we have {1, 2, 4, 6, 10}.
- From set Q, we have {2, 3, 6, 9}.
- Combining the elements from both sets, we get the union PUQ = {1, 2, 4, 6, 9, 10}.
(b) To find PNQ:
PNQ represents the intersection of sets P and Q, which includes all elements that are present in both set P and set Q.
PNQ = {2, 6}
Explanation:
- From set P, we have {1, 2, 4, 6, 10}.
- From set Q, we have {2, 3, 6, 9}.
- The common elements in both sets are 2 and 6, so the intersection PNQ = {2, 6}.
(c) To find p'UQ':
p' represents the complement of set P, which includes all elements that are not present in set P. Q' represents the complement of set Q, which includes all elements that are not present in set Q. So, p'UQ' represents the union of the complement of set P and the complement of set Q.
p' = {3, 5, 7, 8, 9}
Q' = {1, 4, 5, 7, 8, 10}
p'UQ' = {1, 3, 4, 5, 7, 8, 9, 10}
Explanation:
- The complement of set P, p' = {3, 5, 7, 8, 9} (elements not present in set P).
- The complement of set Q, Q' = {1, 4, 5, 7, 8, 10} (elements not present in set Q).
- Combining the elements from both sets, we get the union p'UQ' = {1, 3, 4, 5, 7, 8, 9, 10}.
(d) To find n[pUQ]:
n represents the complement of a set. So, n[pUQ] represents the complement of the union of sets P and Q, which includes all elements that are not present in the union of sets P and Q.
n[pUQ] = {3, 5, 7, 8}
Explanation:
- The union of sets P and Q, [pUQ] = {1, 2, 4, 6, 9, 10}.
- The complement of [pUQ], n[pUQ] = {3, 5, 7, 8} (elements not present in the union [pUQ]).
(e) To find p'nQ':
p' represents the complement of set P, and Q' represents the complement of set Q. So, p'nQ' represents the intersection of the complement of set P and the complement of set Q.
p' = {3, 5, 7, 8, 9}
Q' = {1, 4, 5, 7, 8, 10}
p'nQ' = {5, 7, 8}
Explanation:
- The complement of set P, p' = {3, 5, 7, 8, 9} (elements not present in set P).
- The complement of set Q, Q' = {1, 4, 5, 7, 8, 10} (elements not present in set Q).
- The common elements in both sets are 5, 7, and 8, so the intersection p'nQ' = {5, 7, 8}.
To find the answers to these set operations, we'll use some basic set theory principles. Here's how you can solve each part:
(a) PUQ - This represents the union of sets P and Q. To find the union, you simply combine all the elements from both sets without repeating any elements.
PUQ = {1, 2, 3, 4, 6, 9, 10}
(b) PNQ - This represents the intersection of sets P and Q. To find the intersection, you need to identify the common elements in both sets.
PNQ = {2, 6}
(c) p'UQ' - This represents the complement of set P combined with the complement of set Q. To find the complement of a set, you subtract the set elements from the universal set U.
p' = U - P = {3,5,7,8,9}
Q' = U - Q = {1,4,5,7,8,10}
p'UQ' = {3, 5, 7, 8, 9, 1, 4, 5, 7, 8, 10} [Note: Repeated elements are not considered in sets.]
(d) n[pUQ] - This represents the intersection of the complement of set P with the union of sets P and Q.
[pUQ] = {1, 2, 4, 6, 10}
P' = U - P = {3,5,7,8,9}
n[pUQ] = {3,5,7,8,9}
(e) p'nQ' - This represents the intersection of set P's complement with set Q's complement.
p' = {3,5,7,8,9}
Q' = {1,4,5,7,8,10}
p'nQ' = {5, 7, 8}