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Chapter 2: Problem 17

In Exercises 13-24, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\\B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(A^{\prime} \cap\left(B \cup C^{\prime}\right)\)

Short Answer

Expert verified
The set resulting from the given operation \(A^{\prime} \cap\left(B \cup C^{\prime}\right)\) is \(\{\mathrm{b}\}\).

Step by step solution

01

Find the Complement of A and C

The complementary set \(A^{\prime}\) is such a set that contains all those elements which are in universal set U and not in A. Calculating it: \(A^{\prime} = U - A = \{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\}\). Similarly, complement of set C, \(C^{\prime} = U - C = \{\mathrm{a}, \mathrm{g}, \mathrm{h}\}\).
02

Union of B and Complement of C

The union of set B and \(C^{\prime}\) results in a set that consists of all the elements that are in B or in \(C^{\prime}\), or in both. Calculating it: \(B \cup C^{\prime} = \{\mathrm{a}, \mathrm{b}, \mathrm{g}, \mathrm{h}\}\).
03

Intersection of Complement of A and Union of B and C

The intersection of set \(A^{\prime}\) and \(B \cup C^{\prime}\) results in a set that consists of all the elements that are common to both \(A^{\prime}\) and \(B \cup C^{\prime}\). Calculating it: \(A^{\prime} \cap (B \cup C^{\prime}) = \{\mathrm{b}\}\).

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Key Concepts

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

Complement of a Set
In set theory, the complement of a set, denoted as \(A'\), involves finding all elements from a universal set \(U\) that are not present in set \(A\). Imagine \(U\) as a basket containing all possible items, and \(A\) as a basket containing a few selected items. The complement would be all items left in \(U\) after removing those in \(A\).
For example, if \(U = \{a, b, c, d, e, f, g, h\}\) and \(A = \{a, g, h\}\), then the complement \(A' = \{b, c, d, e, f\}\). This set of elements represents those not chosen by set \(A\), highlighting how the complement effectively captures everything outside of the selected group.
Union of Sets
The union of sets refers to the combination of all unique elements from two or more sets. It is represented by the symbol \(\cup\). When two sets are united, you gather all elements from both without repeating any.
Consider sets \(B = \{b, g, h\}\) and the complement of \(C\), \(C' = \{a, g, h\}\). The union \(B \cup C' = \{a, b, g, h\}\). This set includes all elements that appear in \(B\), \(C'\), or both.
  • Use union to find the range from multiple criteria.
  • The union operation helps in combining data logically.
In practical terms, a union is like pooling all resources while respecting unique contributions from each set.
Intersection of Sets
Intersection focuses on finding common elements between sets. It is denoted by \(\cap\). When taking the intersection, you list only those elements that are present in all sets being compared.
Using our previous example, consider \(A' = \{b, c, d, e, f\}\) and \(B \cup C' = \{a, b, g, h\}\). Their intersection \(A' \cap (B \cup C') = \{b\}\) reveals the elements that \(A'\) and \(B \cup C'\) share.
  • An intersection identifies overlap or commonality.
  • Useful for finding shared characteristics or criteria.
Think of intersection as selecting only the overlapping areas between two Venn circles.
Universal Set
The universal set, often denoted by \(U\), is a comprehensive collection that includes all possible elements under consideration for a particular discussion or problem. Think of \(U\) as the big circle that encompasses every possible option.
In our examples, \(U = \{a, b, c, d, e, f, g, h\}\). It acts as a reference point to determine the complements and relationships of subsets within it.
  • Universal sets vary depending on context鈥攖hey adapt to the scope of the problem.
  • It serves as the starting lineup from which all subsets are derived.
Utilizing the universal set helps define boundaries for calculations and ensures a complete perspective.

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Most popular questions from this chapter

If you are given four sets, \(A, B, C\), and \(U\), describe what is involved in determining \((A \cup B)^{\prime} \cap C\). Be as specific as possible in your description.

In the August 2005 issue of Consumer Reports, readers suffering from depression reported that alternative treatments were less effective than prescription drugs. Suppose that 550 readers felt better taking prescription drugs, 220 felt better through meditation, and 45 felt better taking St. John's wort. Furthermore, 95 felt better using prescription drugs and meditation, 17 felt better using prescription drugs and St. John's wort, 35 felt better using meditation and St. John's wort, 15 improved using all three treatments, and 150 improved using none of these treatments. (Hypothetical results are partly based on percentages given in Consumer Reports.) a. How many readers suffering from depression were included in the report? Of those included in the report, b. How many felt better using prescription drugs or meditation? c. How many felt better using St. John's wort only? d. How many improved using prescription drugs and meditation, but not St. John's wort? e. How many improved using prescription drugs or St. John's wort, but not meditation? f. How many improved using exactly two of these treatments? g. How many improved using at least one of these treatments?

In Exercises 1-12, let $$ \begin{aligned} U &=\\{1,2,3,4,5,6,7\\} \\ A &=\\{1,3,5,7\\} \\ B &=\\{1,2,3\\} \\ C &=\\{2,3,4,5,6\\} . \end{aligned} $$ Find each of the following sets. \(C^{\prime} \cap\left(A \cup B^{\prime}\right)\)

In Exercises 13-24, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\\B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(A \cup(B \cap C)\)

In Exercises 1-12, let $$ \begin{aligned} U &=\\{1,2,3,4,5,6,7\\} \\ A &=\\{1,3,5,7\\} \\ B &=\\{1,2,3\\} \\ C &=\\{2,3,4,5,6\\} . \end{aligned} $$ Find each of the following sets. \((A \cup B) \cap(A \cup C)\)
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