IDNLearn.com provides a user-friendly platform for finding and sharing knowledge. Discover in-depth and trustworthy answers from our extensive network of knowledgeable professionals.
Sagot :
To solve the given problem, we need to use principles from set theory and properties such as union, intersection, and complement of sets. Let's go step by step.
Given:
- Universal set [tex]\( U \)[/tex] has 23 elements: [tex]\( n(U) = 23 \)[/tex]
- [tex]\( n(A \cap B) = 9 \)[/tex]
- [tex]\( n(A \cap C) = 9 \)[/tex]
- [tex]\( n(B \cap C) = 9 \)[/tex]
- [tex]\( n(A \cap B \cap C) = 4 \)[/tex]
- [tex]\( n(A \cup B \cup C) = 19 \)[/tex]
### Part (a) [tex]\( n(A \cup B) \)[/tex]
Since [tex]\( n(A \cup B \cup C) = 19 \)[/tex], this is already provided and pertains to the union of sets [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex].
To find [tex]\( n(A \cup B) \)[/tex], we use the formula for the union of two sets:
[tex]\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \][/tex]
However, since the original question specifies [tex]\( A \cup B = 19 \)[/tex] directly:
[tex]\[ n(A \cup B) = 19 \][/tex]
### Part (b) [tex]\( n(A' \cup C) \)[/tex]
The complement [tex]\( A' \)[/tex] (also written as [tex]\( A^c \)[/tex]) means the set of elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex]. Here's the formula we will use for the union involving a complement:
[tex]\[ n(A' \cup C) = n(U) - n(A \cap C') = n(U) - n(A) + n(A \cap C) - n(C) \][/tex]
Since [tex]\( n(A \cap B \cap C) = 4 \)[/tex], applying the formula: [tex]\( n(A \cap C) \)[/tex] includes elements in their common intersection [tex]\(n(A \cap B \cap C)\)[/tex], so:
[tex]\[ n(A) + n(C) - n(A \cap C) + n(A' \cap C) = n(U) \][/tex]
We know from De Morgan's laws:
[tex]\[ A' \cup C = (A \cap C') \cup C = U - A \][/tex]
Thus:
[tex]\[ n(A' \cup C) = n(U) - n(A) + n(C) - n(A \cap B \cap C)\][/tex]
Since :
[tex]\[ n(U) = 23 - 19=4\][/tex]
### Part (c) [tex]\( n(A \cap B)' \)[/tex]
The complement of [tex]\( n(A \cap B) \)[/tex] is the total number of elements not in [tex]\( A \cap B \)[/tex], given by:
[tex]\[ n(A \cap B)' = n(U) - n(A \cap B) \][/tex]
Given:
[tex]\[ n(U) = 23 \][/tex]
[tex]\[ n(A \cap B) = 9 \][/tex]
So:
[tex]\[ n(A \cap B)' = 23 - 9 = 14 \][/tex]
To summarize:
[tex]\[ \begin{align*} a) \quad &n(A \cup B) = 19 \\ b) \quad &n(A' \cup C) = 4 \\ c) \quad &n(A \cap B)' = 14 \end{align*} \][/tex]
These are the simplified answers to the given questions.
Given:
- Universal set [tex]\( U \)[/tex] has 23 elements: [tex]\( n(U) = 23 \)[/tex]
- [tex]\( n(A \cap B) = 9 \)[/tex]
- [tex]\( n(A \cap C) = 9 \)[/tex]
- [tex]\( n(B \cap C) = 9 \)[/tex]
- [tex]\( n(A \cap B \cap C) = 4 \)[/tex]
- [tex]\( n(A \cup B \cup C) = 19 \)[/tex]
### Part (a) [tex]\( n(A \cup B) \)[/tex]
Since [tex]\( n(A \cup B \cup C) = 19 \)[/tex], this is already provided and pertains to the union of sets [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex].
To find [tex]\( n(A \cup B) \)[/tex], we use the formula for the union of two sets:
[tex]\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \][/tex]
However, since the original question specifies [tex]\( A \cup B = 19 \)[/tex] directly:
[tex]\[ n(A \cup B) = 19 \][/tex]
### Part (b) [tex]\( n(A' \cup C) \)[/tex]
The complement [tex]\( A' \)[/tex] (also written as [tex]\( A^c \)[/tex]) means the set of elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex]. Here's the formula we will use for the union involving a complement:
[tex]\[ n(A' \cup C) = n(U) - n(A \cap C') = n(U) - n(A) + n(A \cap C) - n(C) \][/tex]
Since [tex]\( n(A \cap B \cap C) = 4 \)[/tex], applying the formula: [tex]\( n(A \cap C) \)[/tex] includes elements in their common intersection [tex]\(n(A \cap B \cap C)\)[/tex], so:
[tex]\[ n(A) + n(C) - n(A \cap C) + n(A' \cap C) = n(U) \][/tex]
We know from De Morgan's laws:
[tex]\[ A' \cup C = (A \cap C') \cup C = U - A \][/tex]
Thus:
[tex]\[ n(A' \cup C) = n(U) - n(A) + n(C) - n(A \cap B \cap C)\][/tex]
Since :
[tex]\[ n(U) = 23 - 19=4\][/tex]
### Part (c) [tex]\( n(A \cap B)' \)[/tex]
The complement of [tex]\( n(A \cap B) \)[/tex] is the total number of elements not in [tex]\( A \cap B \)[/tex], given by:
[tex]\[ n(A \cap B)' = n(U) - n(A \cap B) \][/tex]
Given:
[tex]\[ n(U) = 23 \][/tex]
[tex]\[ n(A \cap B) = 9 \][/tex]
So:
[tex]\[ n(A \cap B)' = 23 - 9 = 14 \][/tex]
To summarize:
[tex]\[ \begin{align*} a) \quad &n(A \cup B) = 19 \\ b) \quad &n(A' \cup C) = 4 \\ c) \quad &n(A \cap B)' = 14 \end{align*} \][/tex]
These are the simplified answers to the given questions.
We are happy to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. IDNLearn.com is your reliable source for accurate answers. Thank you for visiting, and we hope to assist you again.
