Chapter 2

Determine if the given sets are equal. $$\\{x, y, z\\},\\{x, z, y\\}$$

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. Find \(|A|\) if \(|A|=|B|,|A \cup B|=2 a+3 b,\) and \(|A \cap B|=b.\)

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. $$(A-B)-C$$

Identify the set S that is defined recursively. i) \(2 \in S\) ii) \(x, y \in S \rightarrow x \pm y \in S\)

In Exercises \(7-10,\) identify the set S that is defined recursively. $$ \begin{array}{l}{\text { i) } \emptyset \in S} \\ {\text { ii) } x \in X, A \in S \rightarrow\\{\mathrm{x}\\} \cup A \in S}\end{array} $$

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. Find \(|A \cap B|\) if \(|A|=a+b=|B|\) and \(|A \cup B|=2 a+2 b.\)

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. $$ A-(B-C) $$

Define each language \(L\) over the given alphabet recursively. $$\\{0,00,10,100,110,0000,1010, \ldots\\}, \Sigma=\\{0,1\\}$$

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. $$ A \cap B $$

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. Find \(|A \cap B|\) if \(|A|=2 a,|B|=a,\) and \(|A \cup B|=2 a+b.\)

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. $$ (A \cup B)-C $$

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. $$(A \cup B)-C$$

Determine if the given sets are equal. $$\left\\{x | x^{2}=x\right\\},\\{0,1\\}$$

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

Find \(|A \cap B|\) if \(|A|=2 a,|B|=a,\) and \(|A \cup B|=2 a+b\).

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. $$ A \cup B $$

Let \(A\) and \(B\) be finite sets such that \(A \subseteq B,|A|=b,|B|=a+b .\) 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. $$ (A \cap B)-C $$

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

Define each language \(L\) over the given alphabet recursively. $$L=\\{1,11,111,1111,11111, \ldots\\}, \Sigma=\\{0,1\\}$$

Determine if the given sets are equal. $$\\{x,\\{y\\}\\},\\{\\{x\\}, y\\}$$

Define each language \(L\) over the given alphabet recursively. $$L=\left\\{x \in \Sigma^{*} | x=\mathrm{b}^{n} \mathrm{ab}^{n}, n \geq 0\right\\}, \Sigma=\\{\mathrm{a}, \mathrm{b}\\}$$

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. $$ B^{\prime} $$

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

Define each language \(L\) over the given alphabet recursively. The language \(L\) of all palindromes over \(\Sigma=\\{a, b] .\) (A palindrome is a word that reads the same both forwards and backwards. For instance, abba is a palindrome.)

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. $$ A-B $$

Mark each as true or false. $$ \mathrm{b} \subseteq\\{\mathrm{a}, \mathrm{b}, \mathrm{c}\\} $$

Let \(A\) and \(B\) be finite sets such that \(A \subseteq B,|A|=b,|B|=a+b .\) Find the cardinality of each set. \(B-A\)

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

The language \(L\) of all palindromes over \(\Sigma=\\{a, b\\} .\) (A palindrome is a word that reads the same both forwards and backwards. For instance, abba is a palindrome.)

Mark each as true or false. $$\mathbf{b} \subseteq\\{\mathbf{a}, \mathbf{b}, \mathbf{c}\\}$$

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. $$ C-B $$

Let \(A\) and \(B\) be finite sets such that \(A \subseteq B,|A|=b,|B|=a+b .\) Find the cardinality of each set. \(A \cap B\)

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

Define each language \(L\) over the given alphabet recursively. $$\\{b, b b, b b b, b b b b, \ldots\\}, \Sigma=\\{a, b\\}$$

Mark each as true or false. $$\\{x\\} \subseteq\\{x, y, z\\}$$

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. $$ A \oplus B $$

Mark each as true or false. $$ \\{0\\}=\emptyset $$

Let \(A\) and \(B\) be finite disjoint sets, where \(|A|=a,\) and \(|B|=b .\) Find the cardinality of each set. \(A \cup B\)

Define each language \(L\) over the given alphabet recursively. $$\\{b, aba, aabaa, aaabaaa, \dots\\}, \Sigma=\\{a, b\\}$$

Mark each as true or false. $$\\{0\\}=\varnothing$$

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. $$ B \oplus C $$

Let \(A\) and \(B\) be finite disjoint sets, where \(|A|=a,\) and \(|B|=b .\) Find the cardinality of each set. \(A-B\)

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

Define each language \(L\) over the given alphabet recursively. $$\\{\mathrm{a}, \text { aaa, aaaaa, aaaaaaa, } \ldots\\}, \Sigma=\\{\mathrm{a}, \mathrm{b}\\}$$

Mark each as true or false. $$0 \in \varnothing$$

Define each language \(L\) over the given alphabet recursively. $$\\{1,10,11,100,101, \ldots\\}, \Sigma=\\{0,1\\}$$

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. $$ C \oplus A $$

Mark each as true or false. $$ |\boldsymbol{Q}|=0 $$

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