Chapter 2

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

Every set is a subset of itself.

Find the number of positive integers \(\leq 500\) and divisible by: Two or three, but not six.

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

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

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

Mark each as true or false. Every set is a subset of itself.

List the well-formed sequences of parentheses with three pairs of left and right parentheses.

Every nonempty set has at least two subsets.

Find the number of positive integers \(\leq 500\) and divisible by: Neither two, three, nor five.

Mark each as true or false, where \(A, B,\) and \(C\) are arbitrary sets and \(U\) the universal set. $$A-\varnothing=A$$

Mark each as true or false. Every nonempty set has at least two subsets.

Find the number of positive integers \(\leq 1776\) and divisible by: Two, three, or five.

Mark each as true or false, where \(A, B,\) and \(C\) are arbitrary sets and \(U\) the universal set. $$\varnothing-A=-A$$

Using Example \(2.37,\) determine if each is a wff in propositional logic. $$ (p \wedge((\sim(q)) \vee r)) $$

Find the number of positive integers \(\leq 1776\) and divisible by: Two, three, or five, but not six.

Mark each as true or false, where \(A, B,\) and \(C\) are arbitrary sets and \(U\) the universal set. $$\varnothing-\varnothing=0$$

Determine if each is a wff in propositional logic. $$(p \wedge((\sim(q)) \vee r))$$

Find the power set of each set. $$ \emptyset $$

Find the number of positive integers \(\leq 1776\) and divisible by: Two, three, or five, but not \(15 .\)

Mark each as true or false, where \(A, B,\) and \(C\) are arbitrary sets and \(U\) the universal set. $$A-A=0$$

Find the power set of each set. $$\varnothing$$

Using Example \(2.37,\) determine if each is a wff in propositional logic. $$ (((\sim p) \vee q) \wedge(\sim q) \vee(\sim p)) ) $$

Find the power set of each set. $$ \mathrm{~ \\{ a \\} ~} $$

Find the number of positive integers \(\leq 1776\) and divisible by: Two, three, or five, but not \(30 .\)

Determine if each is a wff in propositional logic. $$(((\sim p) \vee q) \wedge(\sim q) \vee(\sim p)))$$

Find the power set of each set. $$\\{\mathrm{a}\\}$$

Using Example \(2.37,\) determine if each is a wff in propositional logic. $$ ((p \vee q) \wedge((\sim(q)) \vee(\sim(r)))) $$

Find the power set of each set. $$ \\{a, b, c | $$

According to a survey among 160 college students, 95 students take a course in English, 72 take a course in French, 67 take a course in German, 35 take a course in English and in French, 37 take a course in French and in German, 40 take a course in German and in English, and 25 take a course in all three languages. Find the number of students in the survey who take a course in: English, but not German.

Determine if each is a wff in propositional logic. $$((p \vee q) \wedge((\sim(q)) \vee(\sim(r))))$$

Find the power set of each set. $$\\{\mathrm{a}, \mathrm{b}, \mathrm{c}\\}$$

Determine if the following recursive definition yields the set \(S\) of legally paired parentheses. If not, find a validly paired sequence that cannot be generated by this definition. i) ()\(\in S\). ii) If \(x \in S,\) then ()\(x,(x), x() \in S\)

According to a survey among 160 college students, 95 students take a course in English, 72 take a course in French, 67 take a course in German, 35 take a course in English and in French, 37 take a course in French and in German, 40 take a course in German and in English, and 25 take a course in all three languages. Find the number of students in the survey who take a course in: English, French, or German.

Mark each as true or false, where \(A, B,\) and \(C\) are arbitrary sets and \(U\) the universal set. $$(A \cup B)^{\prime}=A^{\prime} \cup B^{\prime}$$

Define the set of words \(S\) over an alphabet \(\Sigma\) recursively. Assume \(\lambda \in S\). (Hint: use concatenation.)

In Exercises \(34-37, n\) denotes a positive integer less than \(10 .\) Rewrite each set using the listing method. \(\\{n | n \text { is divisible by } 2\\}\)

According to a survey among 160 college students, 95 students take a course in English, 72 take a course in French, 67 take a course in German, 35 take a course in English and in French, 37 take a course in French and in German, 40 take a course in German and in English, and 25 take a course in all three languages. Find the number of students in the survey who take a course in: English or French, but not German.

Mark each as true or false, where \(A, B,\) and \(C\) are arbitrary sets and \(U\) the universal set. $$\left(\boldsymbol{A}^{\prime}\right)^{\prime}=\boldsymbol{A}$$

\(n\) denotes a positive integer less than \(10 .\) Rewrite each set using the listing method. \(\\{n | n \text { is divisible by } 2\\}\)

In Exercises \(34-37, n\) denotes a positive integer less than \(10 .\) Rewrite each set using the listing method. \(\\{n | n \text { is divisible by } 3\\}\)

According to a survey among 160 college students, 95 students take a course in English, 72 take a course in French, 67 take a course in German, 35 take a course in English and in French, 37 take a course in French and in German, 40 take a course in German and in English, and 25 take a course in all three languages. Find the number of students in the survey who take a course in: English and French, but not German.

Let \(\Sigma\) be an alphabet. Define \(\Sigma^{*}\) recursively. (Hint: use concatenation.)

\(n\) denotes a positive integer less than \(10 .\) Rewrite each set using the listing method. \(\\{n | n \text { is divisible by } 3\\}\)

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

Define the language \(L\) of all binary representations of nonnegative integers recursively.

In Exercises \(34-37, n\) denotes a positive integer less than \(10 .\) Rewrite each set using the listing method. \(\\{n | n \text { is divisible by } 2 \text { and } 3\\}\)

According to a survey among 160 college students, 95 students take a course in English, 72 take a course in French, 67 take a course in German, 35 take a course in English and in French, 37 take a course in French and in German, 40 take a course in German and in English, and 25 take a course in all three languages. Find the number of students in the survey who take a course in: English, but neither French nor German.

Mark each as true or false, where \(A, B,\) and \(C\) are arbitrary sets and \(U\) the universal set. $$A \subseteq A \cup B$$

\(n\) denotes a positive integer less than \(10 .\) Rewrite each set using the listing method. \(\\{n | n \text { is divisible by } 2 \text { and } 3\\}\)