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Chapter 2: Problem 5

Let \(A=\\{a, e, f, g, i\rangle, B=\\{b, d, e, g, h\\}, C=\\{d, e, f, h, i\\},\) and \(U=\\{a, b, \ldots, k\\}\) Find each set. $$ (B \cap C)^{\prime} $$

Short Answer

Expert verified
\((B \cap C)^{\prime} = \{a, b, c, f, g, i, j, k\}\)

Step by step solution

01

1. Find the intersection of sets B and C

To find the intersection of sets B and C, we need to identify the elements present in both sets. We'll compare the sets and list the common elements: \(B = \{b, d, e, g, h\}\) \(C = \{d, e, f, h, i\}\) \(B \cap C = \{d, e, h\}\)
02

2. Find the complement of the intersection relative to set U

Now, we will find the complement of the intersection set relative to the universal set U. This will include all elements in the set U, but not in the intersection set: \(U = \{a, b, \ldots, k\}\) \(B \cap C = \{d, e, h\}\) \((B \cap C)^{\prime} = \{a, b, c, f, g, i, j, k\}\) So, \((B \cap C)^{\prime} = \{a, b, c, f, g, i, j, k\}\).

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Most popular questions from this chapter

In Exercises \(1-6,\) a set \(S\) is defined recursively. Find four elements in each case. $$ \begin{array}{l}{\text { i) } 1 \in S} \\ {\text { ii) } x \in S \rightarrow 2^{x} \in S}\end{array} $$

Using the sets \(A=\\{a, b, e, h\\}, B=\\{b, c, e, f, h |, C=\\{c, d, f, g\\}, \text { and }\) \(U=\\{a, \ldots, h\\},\) find the binary representation of each set. $$ A \cup B $$

In Exercises \(1-6,\) a set \(S\) is defined recursively. Find four elements in each case. $$ \begin{array}{l}{\text { i) } e \in S} \\ {\text { ii) } x \in S \rightarrow e^{x} \in S}\end{array} $$

Using the sets \(A=\\{a, b, e, h\\}, B=\\{b, c, e, f, h |, C=\\{c, d, f, g\\}, \text { and }\) \(U=\\{a, \ldots, h\\},\) find the binary representation of each set. $$ A \cap C^{\prime} $$

The empty set is a subset of every set. (Hint: Consider the implication \(x \in \emptyset \rightarrow x \in A .\) )
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