/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 If \(U=\\{1,2,3,4,5,6,7,8,9\\}, ... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 1

If \(U=\\{1,2,3,4,5,6,7,8,9\\}, A=\\{2,4,6\\}\) \(B=\\{1,3,5,7\\}\) and \(C=\\{2,3,4,7,8\\}\) find the sets (a) \(\overline{A \cup B}\) (b) \(C-A\) (c) \(\bar{C} \cap \bar{B}\)

Short Answer

Expert verified
(a) \(\{8,9\}\), (b) \(\{3,7,8\}\), (c) \(\{6,9\}\)

Step by step solution

01

Determine A Union B

We have sets \(A = \{2,4,6\}\) and \(B = \{1,3,5,7\}\). The union of \(A\) and \(B\) is the set of elements that are in either \(A\) or \(B\) or both. Thus, \(A \cup B = \{1,2,3,4,5,6,7\}\).
02

Find Complement of A Union B

The universal set \(U = \{1,2,3,4,5,6,7,8,9\}\), and we need to find \(\overline{A \cup B}\), which is the set of elements in \(U\) that are not in \(A \cup B\). This gives us \(\overline{A \cup B} = \{8,9\}\).
03

Find Elements in C not in A

Set \(C = \{2,3,4,7,8\}\) and set \(A = \{2,4,6\}\). The difference \(C - A\) is the set of elements that are in \(C\) but not in \(A\). Therefore, \(C - A = \{3,7,8\}\).
04

Determine Complement of C

The complement of set \(C\) relative to \(U\) is \(\bar{C}\). With \(C = \{2,3,4,7,8\}\), \(\bar{C} = \{1,5,6,9\}\) since these are the elements of \(U\) not in \(C\).
05

Determine Complement of B

For set \(B = \{1,3,5,7\}\), its complement \(\bar{B}\) contains elements of \(U\) that are not in \(B\). Hence, \(\bar{B} = \{2,4,6,8,9\}\).
06

Find Intersection of Complement C and Complement B

We found \(\bar{C} = \{1,5,6,9\}\) and \(\bar{B} = \{2,4,6,8,9\}\). The intersection \(\bar{C} \cap \bar{B}\) includes elements common to both. Therefore, \(\bar{C} \cap \bar{B} = \{6,9\}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Union of Sets
The union of two sets, say set \(A\) and set \(B\), is a fundamental set operation where you combine all unique elements from both sets into a new set. In symbols, it's expressed as \(A \cup B\). To compute \(A \cup B\), which means "\(A\) union \(B\)", you take each element from set \(A\) and include each element from set \(B\) as well, ensuring no repetition of elements.
The main idea is that the union combines all distinct elements.
For example, in our exercise, \(A = \{2,4,6\}\) and \(B = \{1,3,5,7\}\).
Thus, the union is:
  • \(A \cup B = \{1,2,3,4,5,6,7\}\)
The elements come from both sets, which may have no overlap.
Complement of a Set
The complement of a set involves determining what elements are not present in a given set when compared to a universal set \(U\). The complement of a set \(A\) is noted as \(\overline{A}\) or \(A'\). This represents all the elements in the universal set \(U\) that are not in \(A\).
It's useful for calculating elements outside specific subsets.
Consider \(U = \{1,2,3,4,5,6,7,8,9\}\) and \(A = \{2,4,6\}\).
The complement of \(A\) in this context is:
  • \(\overline{A} = \{1,3,5,7,8,9\}\)
This involves all items in \(U\) that \(A\) doesn't have.
Set Difference
The set difference between two sets \(C - A\) (also known as relative complement), is a calculation where you identify elements that are in set \(C\) but not in \(A\).
This operation is handy when you need to subtract elements of one set from another.
Given \(C = \{2,3,4,7,8\}\) and \(A = \{2,4,6\}\), you'll determine what's exclusive to \(C\) by excluding any overlapping elements with \(A\).
The resultant difference is:
  • \(C - A = \{3,7,8\}\)
Here, elements 3, 7, and 8 are part of \(C\) but not \(A\).
Intersection of Sets
The intersection of sets focuses on shared elements between two sets. For any two sets \(X\) and \(Y\), the intersection, denoted as \(X \cap Y\), includes only those elements present in both sets.
This operation is useful to identify commonalities.
Given complements \(\overline{C} = \{1,5,6,9\}\) and \(\overline{B} = \{2,4,6,8,9\}\), the task is to find the elements that are in both \(\overline{C}\) and \(\overline{B}\).
The result is:
  • \(\overline{C} \cap \overline{B} = \{6,9\}\)
These are the elements shared by both complements.

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