Chapter 2

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

Determine if each sequence of parentheses is legal. $$(()())$$

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

Mark each as true or false. $$ |\emptyset|=\emptyset $$

Let \(A=\\{b, c\\}, B=\\{x\\},\) and \(C=\\{x, z\\} .\) Find each set. $$ A \times B $$

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. \(A \cap C^{\prime}\)

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

Let \(A=\\{\mathrm{b}, \mathrm{c}\\}, B=\\{\mathrm{x}\\},\) and \(C=\\{\mathrm{x}, \mathrm{z}\\} .\) Find each set. $$A \times B$$

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

Mark each as true or false. $$ \emptyset \subseteq \emptyset $$

Let \(A=\\{b, c\\}, B=\\{x\\},\) and \(C=\\{x, z\\} .\) Find each set. $$ B \times A $$

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

Mark each as true or false. $$\varnothing \subseteq \varnothing$$

Let \(A=\\{\mathrm{b}, \mathrm{c}\\}, B=\\{\mathrm{x}\\},\) and \(C=\\{\mathrm{x}, \mathrm{z}\\} .\) Find each set. $$B \times A$$

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. $$ A \cap(B \cap C) $$

Mark each as true or false. $$ \emptyset \in\\{\varnothing\\} $$

Let \(A=\\{b, c\\}, B=\\{x\\},\) and \(C=\\{x, z\\} .\) Find each set. $$ A \times \emptyset $$

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. \(A \cap(B \cap C)\)

Let \(A=\\{\mathrm{b}, \mathrm{c}\\}, B=\\{\mathrm{x}\\},\) and \(C=\\{\mathrm{x}, \mathrm{z}\\} .\) Find each set. $$A \times \emptyset$$

Determine if each sequence of parentheses is legal. $$(()())()$$

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

A survey conducted recently among 300 adults in Omega City shows 160 like to have their houses painted green, and 140 like them blue. Seventy-five adults like both colors. How many do not like either color?

Let \(A=\\{b, c\\}, B=\\{x\\},\) and \(C=\\{x, z\\} .\) Find each set. $$ A \times B \times \emptyset $$

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. \(A \cup(B \cap C)\)

Let \(A=\\{\mathrm{b}, \mathrm{c}\\}, B=\\{\mathrm{x}\\},\) and \(C=\\{\mathrm{x}, \mathrm{z}\\} .\) Find each set. $$A \times B \times \emptyset$$

Mark each as true or false. $$\\{x | x \neq x\\}=\varnothing$$

The nth Catalan number \(\mathrm{C}_{n},\) named after the Belgian mathematician, Eugene Charles Catalan ( \(1814-1894 ),\) is defined by $$ \mathrm{C}_{n}=\frac{(2 n) !}{n !(n+1) !}, \quad n \geq 0 $$ where \(n !(n \text { factorial) is defined by } n !=n(n-1) \ldots 3 \cdot 2 \cdot 1 \text { and } 0 !=1 .\) Catalan numbers have many interesting applications in computer science. For example, the number of well-formed sequences of \(n\) pairs of left and right parentheses is given by the \(n\) th Catalan number. Compute the number of legally paired sequences with the given pairs of left and right parentheses. Three

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

Mark each as true or false. $$ | x, y \\}=|y, x| $$

Let \(A=\\{b, c\\}, B=\\{x\\},\) and \(C=\\{x, z\\} .\) Find each set. $$ A \times(B \cup C) $$

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. \(A-(B \oplus C)\)

Let \(A=\\{\mathrm{b}, \mathrm{c}\\}, B=\\{\mathrm{x}\\},\) and \(C=\\{\mathrm{x}, \mathrm{z}\\} .\) Find each set. $$A \times(B \cap C)$$

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

A survey was taken to determine the preference between two laundry detergents, Lex and Rex. It was found that 15 people liked Lex only, 10 liked both, 20 liked Rex only, and 5 liked neither of them. How many people were surveyed?

The nth Catalan number \(\mathrm{C}_{n},\) named after the Belgian mathematician, Eugene Charles Catalan ( \(1814-1894 ),\) is defined by $$ \mathrm{C}_{n}=\frac{(2 n) !}{n !(n+1) !}, \quad n \geq 0 $$ where \(n !(n \text { factorial) is defined by } n !=n(n-1) \ldots 3 \cdot 2 \cdot 1 \text { and } 0 !=1 .\) Catalan numbers have many interesting applications in computer science. For example, the number of well-formed sequences of \(n\) pairs of left and right parentheses is given by the \(n\) th Catalan number. Compute the number of legally paired sequences with the given pairs of left and right parentheses. Four

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

Find the number of positive integers \(\leq 500\) and divisible by: Two or three.

Let \(A=\\{b, c\\}, B=\\{x\\},\) and \(C=\\{x, z\\} .\) Find each set. $$ A \times(B \cap C) $$

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. \((A \oplus B)-C\)

The \(n\) th Catalan number \(\mathrm{C}_{n},\) named after the Belgian mathematician, Eugene Charles Catalan (1814-1894), is defined by $$\mathrm{C}_{n}=\frac{(2 n) !}{n !(n+1) !}, \quad n \geq 0$$ where \(n !(n \text { factorial) is defined by } n !=n(n-1) \ldots 3 \cdot 2 \cdot 1 \text { and } 0 !=1\) Catalan numbers have many interesting applications in computer science. For example, the number of well-formed sequences of \(n\) pairs of left and right parentheses is given by the \(n\) th Catalan number. Compute the number of legally paired sequences with the given pairs of left and right parentheses. $$Four$$

Let \(A=\\{\mathrm{b}, \mathrm{c}\\}, B=\\{\mathrm{x}\\},\) and \(C=\\{\mathrm{x}, \mathrm{z}\\} .\) Find each set. $$A \times C \times B$$

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

The nth Catalan number \(\mathrm{C}_{n},\) named after the Belgian mathematician, Eugene Charles Catalan ( \(1814-1894 ),\) is defined by $$ \mathrm{C}_{n}=\frac{(2 n) !}{n !(n+1) !}, \quad n \geq 0 $$ where \(n !(n \text { factorial) is defined by } n !=n(n-1) \ldots 3 \cdot 2 \cdot 1 \text { and } 0 !=1 .\) Catalan numbers have many interesting applications in computer science. For example, the number of well-formed sequences of \(n\) pairs of left and right parentheses is given by the \(n\) th Catalan number. Compute the number of legally paired sequences with the given pairs of left and right parentheses. Five

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. $$ A \oplus(B \oplus C) $$

0 is a subset of every set.

Find the number of positive integers \(\leq 500\) and divisible by: Two, three, or five.

Let \(A=\\{b, c\\}, B=\\{x\\},\) and \(C=\\{x, z\\} .\) Find each set. $$ A \times B \times C $$

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. \(A \oplus(B \oplus C)\)

Mark each as true or false. \(\varnothing\) is a subset of every set.

The nth Catalan number \(\mathrm{C}_{n},\) named after the Belgian mathematician, Eugene Charles Catalan ( \(1814-1894 ),\) is defined by $$ \mathrm{C}_{n}=\frac{(2 n) !}{n !(n+1) !}, \quad n \geq 0 $$ where \(n !(n \text { factorial) is defined by } n !=n(n-1) \ldots 3 \cdot 2 \cdot 1 \text { and } 0 !=1 .\) Catalan numbers have many interesting applications in computer science. For example, the number of well-formed sequences of \(n\) pairs of left and right parentheses is given by the \(n\) th Catalan number. Compute the number of legally paired sequences with the given pairs of left and right parentheses. Six