Discrete Mathematics With Applications
5th Edition
ISBN: 9781337694193
Author: EPP, Susanna S.
Publisher: Cengage Learning,
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Chapter 2.4, Problem 17ES
To determine
To construct:
The circuit of the given Boolean expression
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In this exercise, we will investigate a technique to prove that a language is notregular. This tool is called the pumping lemma.The pumping lemma says that if M = (S, I, f, s0, F ) is a DFA with p states (i.e., p = |S|) and if the wordw is in L(M ) (the language generated by M ) and w has length greater than or equal to p, then w may bedivided into three pieces, w = xyz, satisfying the following conditions:1. For each i ∈ N, xy^i z ∈ L(M ).2. |y| > 0 (i.e., y contains at least one character).3. |xy| ≤ p (i.e., the string xy has at most p characters).
Use the pumping lemma to show the following language is not regular (HINT: Use proof by contradictionto assume the language is regular and apply the pumping lemma to the language):L = {0^k1^k | k ∈ N}
A prefix of length ℓ of some word w are the first ℓ characters (in order) of w.1. Construct a context-free grammar for the language: L = {w ∈ {a, b}∗ | every prefix of w has at least as many a’s as b’s}2. Explain why every word generated by your context-free grammar (in Part 1) is contained in L. Then,prove via induction that every w ∈ L is produced by your context-free grammar.
Consider a simplified version of American football where on any possession ateam can earn 0, 3 or 7 points. What is the smallest number n0 of points such that for all n ≥ n0 and n ∈ Na team could earn n points. You must prove that your answer is correct via induction (HINT: Don’t forgetto show that n0 is the smallest number above which any number of points is reachable).
Chapter 2 Solutions
Discrete Mathematics With Applications
Ch. 2.1 - An and statement is true when, and only when, both...Ch. 2.1 - An or statement is false when, and only when, both...Ch. 2.1 - Two statement forms are logically equivalent when,...Ch. 2.1 - De Morgan’s laws say (1) that the negation of an...Ch. 2.1 - A tautology is a statement that is always _____.Ch. 2.1 - A contradiction is a statement that is always...Ch. 2.1 - In eachof 1—4 represent the common form of each...Ch. 2.1 - In each of 1-4 represent the common form of each...Ch. 2.1 - In each of 1—4 represent the common form of each...Ch. 2.1 - In each of 1—4 represent the common form of each...
Ch. 2.1 - Indicate which of the following sentences are...Ch. 2.1 - Write the statements in 6-9 in symbolic form using...Ch. 2.1 - Write the statements in 6-9 in symbolic form using...Ch. 2.1 - Write the statements in 6-9 n symbolic form using...Ch. 2.1 - Write the statements in 6-9 in symbolic form using...Ch. 2.1 - Let p be the statement "DATAENDFLAG is off," q the...Ch. 2.1 - In the following sentence, is the word or used in...Ch. 2.1 - Write truth tables for the statement forms in...Ch. 2.1 - Write truth tables for the statement forms in...Ch. 2.1 - Write truth tables for the statement forms in...Ch. 2.1 - Write truth tables for the statement forms in...Ch. 2.1 - Determine whether the statement forms in 16—24 are...Ch. 2.1 - Determine whether the statement forms in 16-24 are...Ch. 2.1 - Determine whether the statement forms in 16—24 are...Ch. 2.1 - Determine whether the statement forms in 16—24 are...Ch. 2.1 - Determine whether the statement forms in 16—24 are...Ch. 2.1 - Determine whether the statement forms in 16-24 are...Ch. 2.1 - Determine whether the statement forms in 16-24 are...Ch. 2.1 - Determine whether the statement forms in 16-24 are...Ch. 2.1 - Determine whether the statement forms in 16-24 are...Ch. 2.1 - Use De Morgan’s laws to write negations for the...Ch. 2.1 - Use De Morgan’s laws to write negations for the...Ch. 2.1 - Use De Morgan’s laws to write negations for the...Ch. 2.1 - Use De Morgan’s laws to write negations for the...Ch. 2.1 - Use De Morgan’s laws to write negations for the...Ch. 2.1 - Use De Morgan’s laws to write negations for the...Ch. 2.1 - Prob. 31ESCh. 2.1 - Assume x is a particular real number and use De...Ch. 2.1 - Assume x is a particular real number and use De...Ch. 2.1 - Assume x is a particular real number and use De...Ch. 2.1 - Assume x is a particular real number and use De...Ch. 2.1 - Assume x is a particular real number and use De...Ch. 2.1 - Assume x is a particular real number and use De...Ch. 2.1 - In 38 and 39, imagine that num_orders and...Ch. 2.1 - In 38 and 39, imagine that num_orders and...Ch. 2.1 - Use truth to establish which of the statement...Ch. 2.1 - Use truth tables to establish which of the...Ch. 2.1 - Use truth to establish which of the statement...Ch. 2.1 - Use truth tables to establish which of the...Ch. 2.1 - Recall that axb means that ax and xb . Also ab...Ch. 2.1 - Determine whether the statements in (a) and (b)...Ch. 2.1 - Let the symbol denote exclusive or; so...Ch. 2.1 - In logic and in standard English, a double...Ch. 2.1 - In 48 and 49 below, a logical equivalence is...Ch. 2.1 - In 48 and 49 below, a logical equivalence is...Ch. 2.1 - Use Theorem 2.11 to verify the logical...Ch. 2.1 - Use theorem 2.11 to verify the logical...Ch. 2.1 - Use Theorem 2.11 to verify the logical...Ch. 2.1 - Use Theorem 2.11 to verify the logical...Ch. 2.1 - Use Theorem 2.11 to verify the logical...Ch. 2.2 - An if-then statement is false if, and only if, the...Ch. 2.2 - The negation of “if p then q” is _____Ch. 2.2 - The converse of”if p then q” is _______Ch. 2.2 - The contrapositive of “if p the q” is _________Ch. 2.2 - Prob. 5TYCh. 2.2 - A conditional statement and its contrapositive...Ch. 2.2 - Prob. 7TYCh. 2.2 - “R is a sufficient condition for S” means “if...Ch. 2.2 - “R is a necessary condition for S” means “if...Ch. 2.2 - Prob. 10TYCh. 2.2 - Rewrite the statements in 1-4 in if-then form.Ch. 2.2 - Rewrite the statements in 1-4 in if-then from. I...Ch. 2.2 - Rewrite the statements in 1-4 in if-then form....Ch. 2.2 - Prob. 4ESCh. 2.2 - Construct truth tables for the statements forms in...Ch. 2.2 - Construct truth tables for the statements forms in...Ch. 2.2 - Prob. 7ESCh. 2.2 - Prob. 8ESCh. 2.2 - Construct truth tables for the statements forms in...Ch. 2.2 - Prob. 10ESCh. 2.2 - Prob. 11ESCh. 2.2 - Use the logical equivalence established in Example...Ch. 2.2 - Prob. 13ESCh. 2.2 - Show that the following statement forms are all...Ch. 2.2 - Determine whether the following statement forms...Ch. 2.2 - Prob. 16ESCh. 2.2 - In 16 and 17, write each o the two statements in...Ch. 2.2 - Write each at the following three statements in...Ch. 2.2 - True or false? The negation of “If Sue is Luiz’s...Ch. 2.2 - Write negations for each of the following...Ch. 2.2 - Suppose that p and q are statements so that p ) q...Ch. 2.2 - Write negations for each of the following...Ch. 2.2 - Write negations for each of the following...Ch. 2.2 - Prob. 24ESCh. 2.2 - Prob. 25ESCh. 2.2 - Use truth tables to establish the truth of each...Ch. 2.2 - Prob. 27ESCh. 2.2 - Prob. 28ESCh. 2.2 - If statement forms P and Q are logically...Ch. 2.2 - Prob. 30ESCh. 2.2 - If statement forms P mid Q are logically...Ch. 2.2 - Rewrite each of the statements in 32 and 33 as a...Ch. 2.2 - Prob. 33ESCh. 2.2 - Rewrite the statements in 34 and 35 in if-then...Ch. 2.2 - Rewrite the statements in 34 and 35 en in-then...Ch. 2.2 - Taking the long view on u education, you go to the...Ch. 2.2 - Some prograrnming languages use statements of the...Ch. 2.2 - Some programming languages use statements of the...Ch. 2.2 - Prob. 39ESCh. 2.2 - Prob. 40ESCh. 2.2 - Prob. 41ESCh. 2.2 - Prob. 42ESCh. 2.2 - Use the contrapositive to rewrite the statements...Ch. 2.2 - Prob. 44ESCh. 2.2 - Note that a sufficient condition lot s is r”...Ch. 2.2 - “If compound X is boiling, then its temperature...Ch. 2.2 - In 47— 50(a)use the logical equivalences pq=~pq...Ch. 2.2 - In 47— 50(a)use the logical equivalences pq=~pq...Ch. 2.2 - In 47-50 (a) use the logical equivalences pq=~pq...Ch. 2.2 - In 47-50(a) use the logical equivalences pq=~pq...Ch. 2.2 - Given any statement form, is it possible to find a...Ch. 2.3 - For an argument to be valid means that every...Ch. 2.3 - For an argument to be invalid means that there is...Ch. 2.3 - Prob. 3TYCh. 2.3 - Use modus ponens at modus tollens to fill in the...Ch. 2.3 - Use modus ponens or modus tollens to fill in the...Ch. 2.3 - Use modus ponens or modus tollens to fill in the...Ch. 2.3 - Use modus ponens at modus tollens to fill in the...Ch. 2.3 - Use modus ponens or modus tollens to fill in the...Ch. 2.3 - Use truth tables to determine whether the argument...Ch. 2.3 - Prob. 7ESCh. 2.3 - Use truth tables to determine whether the argument...Ch. 2.3 - Use truth tables to determine whether the argument...Ch. 2.3 - Use truth tables to determine whether the argument...Ch. 2.3 - Use truth tables to determine whether the argument...Ch. 2.3 - Use truth table to show that the following forms...Ch. 2.3 - Use truth tables to show that the argument forms...Ch. 2.3 - Prob. 14ESCh. 2.3 - Prob. 15ESCh. 2.3 - Prob. 16ESCh. 2.3 - Prob. 17ESCh. 2.3 - Use truth table to show that the argument forms...Ch. 2.3 - Prob. 19ESCh. 2.3 - Prob. 20ESCh. 2.3 - Prob. 21ESCh. 2.3 - Prob. 22ESCh. 2.3 - Use symbols to write the logical form of each...Ch. 2.3 - Some of the argurnents in 24-32 are valid, whereas...Ch. 2.3 - Prob. 25ESCh. 2.3 - Some at the arguments in 24—32 are valid, whereas...Ch. 2.3 - Prob. 27ESCh. 2.3 - Some of the argents in 24-32 are valid. wherere as...Ch. 2.3 - Some of the arguments in 24-32 are valid, whereas...Ch. 2.3 - Some of the arguments in 24-32 are valid, whereas...Ch. 2.3 - Some of the arguments in 24-32 are valis, whereas...Ch. 2.3 - Some of the arguments in 24-32 are valid, whereas...Ch. 2.3 - Give an example (other then Example 2.3.11) of a...Ch. 2.3 - Give an example (other than Example 2.3.12) of an...Ch. 2.3 - Prob. 35ESCh. 2.3 - Given the following information about a computer...Ch. 2.3 - In the back of an old cupboard you discusser a...Ch. 2.3 - Prob. 38ESCh. 2.3 - The famous detective Percule Hoirot was called in...Ch. 2.3 - Prob. 40ESCh. 2.3 - In 41—44 a set a pren.sei and a conclusion arc...Ch. 2.3 - In 41-44 a set premises and a conclusion are...Ch. 2.3 - In 41-44 a set premises and a conclusion are...Ch. 2.3 - In 41-44 a wt o premises and a conclusion are...Ch. 2.4 - The input/output table for a digital logic circuit...Ch. 2.4 - The Boolean expression that corresponds to a...Ch. 2.4 - Prob. 3TYCh. 2.4 - Prob. 4TYCh. 2.4 - Prob. 5TYCh. 2.4 - Prob. 6TYCh. 2.4 - Prob. 1ESCh. 2.4 - Give the output signals for the circuits in 1—4 if...Ch. 2.4 - Give the output signals for the circuits in 1—4 if...Ch. 2.4 - Give the output signals for the circuits in 1-4 if...Ch. 2.4 - Prob. 5ESCh. 2.4 - Prob. 6ESCh. 2.4 - Prob. 7ESCh. 2.4 - In 5-8, write an input/output table for the...Ch. 2.4 - Prob. 9ESCh. 2.4 - In 9-12, find the Boolean expression that...Ch. 2.4 - Prob. 11ESCh. 2.4 - In 9-12, find the Boolean expression that...Ch. 2.4 - Prob. 13ESCh. 2.4 - Construct circuits for the Boolean expressions in...Ch. 2.4 - Prob. 15ESCh. 2.4 - Prob. 16ESCh. 2.4 - Prob. 17ESCh. 2.4 - For each of the tables in 18-21, construct (a) a...Ch. 2.4 - For each of the tables in 18-21, construct (a) a...Ch. 2.4 - For each of the tables in 18-21, construct (a) a...Ch. 2.4 - For each of the tables in 18-21, construct (a) a...Ch. 2.4 - Design a circuit to take input signals P,Q, and R...Ch. 2.4 - Design a circuit to take input signals P,Q, and R...Ch. 2.4 - The light in a classroom are controlled by two...Ch. 2.4 - An alarm system has three different control panels...Ch. 2.4 - Use the properties listed in Thearem 2.1.1 to to...Ch. 2.4 - Use the properties listed in Theorem 2.1.1 to show...Ch. 2.4 - Use the properties kited in Theorem 2.1.1 to show...Ch. 2.4 - Prob. 29ESCh. 2.4 - For the circuits corresponding to the Boolean...Ch. 2.4 - Prob. 31ESCh. 2.4 - The Boolean expression for the circuit in Example...Ch. 2.4 - Show that for the Sheffer stroke |, PQ(PQ)(PQ)....Ch. 2.4 - Show that the following logical equivalences hold...Ch. 2.5 - To represent a nonnegative integer in binary...Ch. 2.5 - Prob. 2TYCh. 2.5 - Prob. 3TYCh. 2.5 - Prob. 4TYCh. 2.5 - Prob. 5TYCh. 2.5 - Prob. 6TYCh. 2.5 - Prob. 7TYCh. 2.5 - Prob. 8TYCh. 2.5 - Prob. 9TYCh. 2.5 - Represent the decimal integers in 1-6 in binary...Ch. 2.5 - Represent the decimal integers in 1-6 in binary...Ch. 2.5 - Prob. 3ESCh. 2.5 - Prob. 4ESCh. 2.5 - Prob. 5ESCh. 2.5 - Prob. 6ESCh. 2.5 - Represent the integers in 7-12 in decimal...Ch. 2.5 - Prob. 8ESCh. 2.5 - Prob. 9ESCh. 2.5 - Represent the integers in 7—12 in decimal...Ch. 2.5 - Prob. 11ESCh. 2.5 - Represent the integers in 7—12 in decimal...Ch. 2.5 - Perform the arithmetic in 13-20 using binary...Ch. 2.5 - Prob. 14ESCh. 2.5 - Prob. 15ESCh. 2.5 - Prob. 16ESCh. 2.5 - Prob. 17ESCh. 2.5 - Prob. 18ESCh. 2.5 - Prob. 19ESCh. 2.5 - Prob. 20ESCh. 2.5 - Give the output singals S and T for the circuit...Ch. 2.5 - Add 111111112+12 and convert the result to decimal...Ch. 2.5 - Prob. 23ESCh. 2.5 - Prob. 24ESCh. 2.5 - Prob. 25ESCh. 2.5 - Prob. 26ESCh. 2.5 - Prob. 27ESCh. 2.5 - Prob. 28ESCh. 2.5 - Prob. 29ESCh. 2.5 - Prob. 30ESCh. 2.5 - Prob. 31ESCh. 2.5 - Prob. 32ESCh. 2.5 - Use 8-bit two’s complements to compute the surms...Ch. 2.5 - Prob. 34ESCh. 2.5 - Prob. 35ESCh. 2.5 - Prob. 36ESCh. 2.5 - Prob. 37ESCh. 2.5 - Prob. 38ESCh. 2.5 - Prob. 39ESCh. 2.5 - Convert the integers in 38-40 from hexadecimal to...Ch. 2.5 - Prob. 41ESCh. 2.5 - Prob. 42ESCh. 2.5 - Convert the integers in 41-43 from hexadecimal to...Ch. 2.5 - Prob. 44ESCh. 2.5 - Prob. 45ESCh. 2.5 - Prob. 46ESCh. 2.5 - Prob. 47ES
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