Chapter 2

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

Rewrite each set using the listing method. The set of months that begin with the letter A.

Find the cardinality of each set. The set of letters of the English alphabet.

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. $$ C^\prime $$

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

Using the universal set \(U=\\{\mathrm{a}, \ldots, \mathrm{h}\\},\) represent each set as an 8 -bit word. \(\\{a, c, e, g\\}\)

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 universal set \(U=\\{a, \ldots, h |, \text { represent each set as an } 8 \text { -bit word. }\) $$\\{\mathrm{b}, \mathrm{d}, \mathrm{f}\\}$$

Rewrite each set using the listing method. The set of letters of the word GOOGOL.

Find the cardinality of each set. The set of letters of the word TWEEDLEDEE.

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. $$ B \cap C^{\prime} $$

Using the universal set \(U=\\{\mathrm{a}, \ldots, \mathrm{h}\\},\) represent each set as an 8 -bit word. \(\\{b, d, f\\}\)

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

A set \(S\) is defined recursively. Find four elements in each case. i) \(1 \in S\) ii) \(x \in S \rightarrow 2^{x} \in S\)

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 universal set \(U=\\{a, \ldots, h |, \text { represent each set as an } 8 \text { -bit word. }\) $$\\{a, e, f, g, h\\}$$

Rewrite each set using the listing method. The set of months with exactly 31 days.

Find the cardinality of each set. The set of months of the year with 31 days.

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. $$ C \cap A^{\prime} $$

Using the universal set \(U=\\{\mathrm{a}, \ldots, \mathrm{h}\\},\) represent each set as an 8 -bit word. \(\\{a, e, f, g, h\\}\)

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

In Exercises \(1-6,\) a set \(S\) is defined recursively. Find four elements in each case. $$ \begin{array}{l}{\text { i) } 3 \in S} \\ {\text { ii) } x \in S \rightarrow \lg x \in S^{\dagger}}\end{array} $$

Using the universal set \(U=\\{a, \ldots, h |, \text { represent each set as an } 8 \text { -bit word. }\) $$\emptyset$$

Rewrite each set using the listing method. The set of solutions of the equation \(x^{2}-5 x+6=0\)

Find the cardinality of each set. The set of identifiers in Java that begin with 3.

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. $$ (A \cup B)^{\prime} $$

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

Using the universal set \(U=\\{\mathrm{a}, \ldots, \mathrm{h}\\},\) represent each set as an 8 -bit word. \(\emptyset\)

Rewrite each set using the set-builder notation. The set of integers between 0 and \(5 .\)

Let \(A\) and \(B\) be two sets such that \(|A|=2 a-b,|B|=2 a,|A \cap B|=a-b\) and \(|U|=3 a+2 b .\) Find the cardinality of each set. \(A \cup B\)

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. $$ (B \cap C)^{\prime} $$

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

Use Algorithm 2.1 to find the subset of the set \(\left\\{s_{0}, s_{1}, s_{2}, s_{3}\right\\}\) that follows the given subset. \(\left\\{s_{0}, s_{3}\right\\}\)

Rewrite each set using the set-builder notation. The set of January, February, May, and July.

Let \(A\) and \(B\) be two sets such that \(|A|=2 a-b,|B|=2 a,|A \cap B|=a-b\) and \(|U|=3 a+2 b .\) Find the cardinality of each set. \(A-B\)

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. $$ \left(A \cup C^{\prime}\right)^{\prime} $$

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

In Exercises \(7-10,\) identify the set S that is defined recursively. $$ \begin{array}{l}{\text { i) } 1 \in S} \\ {\text { ii) } x, y \in S \rightarrow x+y \in S}\end{array} $$

Use Algorithm 2.1 to find the subset of the set \(\left\\{s_{0}, s_{1}, s_{2}, s_{3}\right\\}\) that follows the given subset. \(\left\\{s_{2}, s_{3}\right\\}\)

Rewrite each set using the set-builder notation. The set of all members of the United Nations.

Let \(A\) and \(B\) be two sets such that \(|A|=2 a-b,|B|=2 a,|A \cap B|=a-b\) and \(|U|=3 a+2 b .\) Find the cardinality of each set. \(B^{\prime}\)

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. $$ (B \cap C)^{\prime} $$

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

Identify the set S that is defined recursively. i) \(1 \in S\) ii) \(x, y \in S \rightarrow x+y \in S\)

In Exercises \(7-10,\) identify the set S that is defined recursively. $$ \begin{array}{l}{\text { i) } 1 \in S} \\ {\text { ii) } x, y \in S \rightarrow x \pm y \in S}\end{array} $$

Let \(A\) and \(B\) be two sets such that \(|A|=2 a-b,|B|=2 a,|A \cap B|=a-b\) and \(|U|=3 a+2 b .\) Find the cardinality of each set. \(A-A^{\prime}\)

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. $$ A \oplus B $$

Rewrite each set using the set-builder notation. {Asia, Australia, Antarctica}

Identify the set S that is defined recursively. i) \(1 \in S\) ii) \(x, y \in S \rightarrow x \pm y \in S\)

In Exercises \(7-10,\) identify the set S that is defined recursively. $$ \begin{array}{l}{\text { i) } 2 \in S} \\ {\text { ii) } x, y \in S \rightarrow x \pm y \in S}\end{array} $$