In each of 14 represent the common form of each argument using letters to stand for component sentences, and fill in the blanks so that the argument in part (b) has the same logical form as the argument in part (a).
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Table of Contents
1.1
Variables
1.1
Variables
1.2
The Language of Sets
1.2
The Language of Sets
1.3
The Language of Relations and Functions
1.3
The Language of Relations and Functions
2.1
Logical Form and Logical Equivalence
2.1
Logical Form and Logical Equivalence
2.2
Conditional Statements
2.2
Conditional Statements
2.3
Valid and Invalid Arguments
2.3
Valid and Invalid Arguments
2.4
Application: Digital Logic Circuits
2.4
Application: Digital Logic Circuits
2.5
Application: Number Systems and Circuits for Addition
2.5
Application: Number Systems and Circuits for Addition
3.1
Predicates and Quantified Statements I
3.2
Predicates and Quantified Statements II
3.2
Predicates and Quantified Statements II
3.3
Statements with Multiple Quantifiers
3.3
Statements with Multiple Quantifiers
3.4
Arguments with Quantified Statements
3.4
Arguments with Quantified Statements
4.1
Direct Proof and Counterexample I: Introduction
4.1
Direct Proof and Counterexample I: Introduction
4.2
Direct Proof and Counterexample II: Rational Numbers
4.2
Direct Proof and Counterexample II: Rational Numbers
4.3
Direct Proof and Counterexample III: Divisibility
4.3
Direct Proof and Counterexample III: Divisibility
4.4
Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem
4.4
Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem
4.5
Direct Proof and Counterexample V: Floor and Ceiling
4.5
Direct Proof and Counterexample V: Floor and Ceiling
4.6
Indirect Argument: Contradiction and Contraposition
4.6
Indirect Argument: Contradiction and Contraposition
4.7
Indirect Argument: Two Classical Theorems
4.7
Indirect Argument: Two Classical Theorems
4.8
Application: Algorithms
4.8
Application: Algorithms
5.1
Sequences
5.1
Sequences
5.2
Mathematical Induction I
5.2
Mathematical Induction I
5.3
Mathematical Induction II
5.3
Mathematical Induction II
5.4
Strong Mathematical Induction and the Well-Ordering Principle for the Integers
5.4
Strong Mathematical Induction and the Well-Ordering Principle for the Integers
5.5
Application: Correctness of Algorithms
5.5
Application: Correctness of Algorithms
5.6
Defining Sequences Recursively
5.6
Defining Sequences Recursively
5.7
Solving Recurrence Relations by Iteration
5.7
Solving Recurrence Relations by Iteration
5.8
Second-Order Linear Homogenous Recurrence Relations with Constant Coefficients
5.8
Second-Order Linear Homogenous Recurrence Relations with Constant Coefficients
5.9
General Recursive Definitions and Structural Induction
5.9
General Recursive Definitions and Structural Induction
6.1
Set Theory: Definitions and the Element Method of Proof
6.1
Set Theory: Definitions and the Element Method of Proof
6.2
Properties of Sets
6.2
Properties of Sets
6.3
Disproofs, Algebraic Proofs, and Boolean Algebras
6.3
Disproofs, Algebraic Proofs, and Boolean Algebras
6.4
Boolean Algebras, Russell’s Paradox, and the Halting Problem
6.4
Boolean Algebras, Russell’s Paradox, and the Halting Problem
7.1
Functions Defined on General Sets
7.1
Functions Defined on General Sets
7.2
One-to-One and Onto, Inverse Functions
7.2
One-to-One and Onto, Inverse Functions
7.3
Composition of Functions
7.3
Composition of Functions
7.4
Cardinality with Applications to Computability
7.4
Cardinality with Applications to Computability
8.1
Relations on Sets
8.1
Relations on Sets
8.2
Reflexivity, Symmetry, and Transitivity
8.2
Reflexivity, Symmetry, and Transitivity
8.3
Equivalence Relations
8.3
Equivalence Relations
8.4
Modular Arithmetic with Applications to Cryptography
8.4
Modular Arithmetic with Applications to Cryptography
8.5
Partial Order Relations
8.5
Partial Order Relations
9.1
Introduction
9.1
Introduction
9.2
Possibility Trees and the Multiplication Rule
9.2
Possibility Trees and the Multiplication Rule
9.3
Counting Elements of Disjoint Sets: The Addition Rule
9.3
Counting Elements of Disjoint Sets: The Addition Rule
9.4
The Pigeonhole Principle
9.4
The Pigeonhole Principle
9.5
Counting Subsets of a Set: Combinations
9.5
Counting Subsets of a Set: Combinations
9.6
r-Combinations with Repetition Allowed
9.6
r-Combinations with Repetition Allowed
9.7
Pascal’s Formula and the Binomial Theorem
9.7
Pascal’s Formula and the Binomial Theorem
9.8
Probability Axioms and Expected Value
9.8
Probability Axioms and Expected Value
9.9
Conditional Probability, Bayes’ Formula, and Independent Events
9.9
Conditional Probability, Bayes’ Formula, and Independent Events
10.1
Graphs: Definitions and Basic Properties
10.1
Graphs: Definitions and Basic Properties
10.2
Trails, Paths, and Circuits
10.2
Trails, Paths, and Circuits
10.3
Matrix Representations of Graphs
10.3
Matrix Representations of Graphs
10.4
Isomorphisms of Graphs
10.4
Isomorphisms of Graphs
10.5
Trees
10.5
Trees
10.6
Rooted Trees
10.6
Rooted Trees
10.7
Spanning Trees and Shortest Paths
10.7
Spanning Trees and Shortest Paths
11.1
Real-Valued Functions of a Real Variable and Their Graphs
11.1
Real-Valued Functions of a Real Variable and Their Graphs
11.2
Notations
11.2
Notations
11.3
Application: Analysis of Algorithm Efficiency I
11.3
Application: Analysis of Algorithm Efficiency I
11.4
Exponential and Logarithmic Functions: Graphs and Orders
11.4
Exponential and Logarithmic Functions: Graphs and Orders
11.5
Application: Analysis of Algorithm Efficiency II
11.5
Application: Analysis of Algorithm Efficiency II
12.1
Formal Languages and Regular Expressions
12.1
Formal Languages and Regular Expressions
12.2
Finite-State Automata
12.2
Finite-State Automata
12.3
Simplifying Finite-State Automata
12.3
Simplifying Finite-State Automata
Textbook Solutions for Discrete Mathematics with Applications
Chapter 2.1 Problem 16
Question
Determine whether the statement forms in 1624 are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence.
Solution
The first step in solving 2.1 problem number 16 trying to solve the problem we have to refer to the textbook question: Determine whether the statement forms in 1624 are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence.
From the textbook chapter Logical Form and Logical Equivalence you will find a few key concepts needed to solve this.
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full solution
full solution
Title
Discrete Mathematics with Applications 4
Author
Susanna S. Epp
ISBN
9780495391326
Solution: Determine whether the statement forms in 1624 are logically equivalent. In
Chapter 2.1 textbook questions
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Chapter 2: Problem 1 Discrete Mathematics with Applications 4
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Chapter 2: Problem 2 Discrete Mathematics with Applications 4
In each of 14 represent the common form of each argument using letters to stand for component sentences, and fill in the blanks so that the argument in part (b) has the same logical form as the argument in part (a).
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Chapter 2: Problem 3 Discrete Mathematics with Applications 4
In each of 14 represent the common form of each argument using letters to stand for component sentences, and fill in the blanks so that the argument in part (b) has the same logical form as the argument in part (a).
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Chapter 2: Problem 4 Discrete Mathematics with Applications 4
In each of 14 represent the common form of each argument using letters to stand for component sentences, and fill in the blanks so that the argument in part (b) has the same logical form as the argument in part (a).
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Chapter 2: Problem 5 Discrete Mathematics with Applications 4
Indicate which of the following sentences are statements. a. 1,024 is the smallest four-digit number that is a perfect square. b. She is a mathematics major. c. 128 = 26 d. x = 26
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Chapter 2: Problem 6 Discrete Mathematics with Applications 4
Write the statements in 69 in symbolic form using the symbols , , and and the indicated letters to represent component statements.
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Chapter 2: Problem 7 Discrete Mathematics with Applications 4
Write the statements in 69 in symbolic form using the symbols , , and and the indicated letters to represent component statements.
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Chapter 2: Problem 8 Discrete Mathematics with Applications 4
Write the statements in 69 in symbolic form using the symbols , , and and the indicated letters to represent component statements.
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Chapter 2: Problem 9 Discrete Mathematics with Applications 4
Write the statements in 69 in symbolic form using the symbols , , and and the indicated letters to represent component statements.
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Chapter 2: Problem 10 Discrete Mathematics with Applications 4
Let p be the statement DATAENDFLAG is off, q the statement ERROR equals 0, and r the statement SUM is less than 1,000. Express the following sentences in symbolic notation. a. DATAENDFLAG is off, ERROR equals 0, and SUM is less than 1,000. b. DATAENDFLAG is off but ERROR is not equal to 0. c. DATAENDFLAG is off; however, ERROR is not 0 or SUM is greater than or equal to 1,000. d. DATAENDFLAG is on and ERROR equals 0 but SUM is greater than or equal to 1,000. e. Either DATAENDFLAG is on or it is the case that both ERROR equals 0 and SUM is less than 1,000.
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Chapter 2: Problem 11 Discrete Mathematics with Applications 4
Let p be the statement DATAENDFLAG is off, q the statement ERROR equals 0, and r the statement SUM is less than 1,000. Express the following sentences in symbolic notation. a. DATAENDFLAG is off, ERROR equals 0, and SUM is less than 1,000. b. DATAENDFLAG is off but ERROR is not equal to 0. c. DATAENDFLAG is off; however, ERROR is not 0 or SUM is greater than or equal to 1,000. d. DATAENDFLAG is on and ERROR equals 0 but SUM is greater than or equal to 1,000. e. Either DATAENDFLAG is on or it is the case that both ERROR equals 0 and SUM is less than 1,000.
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Chapter 2: Problem 12 Discrete Mathematics with Applications 4
Write truth tables for the statement forms in 1215.
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Chapter 2: Problem 13 Discrete Mathematics with Applications 4
Write truth tables for the statement forms in 1215.
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Chapter 2: Problem 14 Discrete Mathematics with Applications 4
Write truth tables for the statement forms in 1215.
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Chapter 2: Problem 15 Discrete Mathematics with Applications 4
Write truth tables for the statement forms in 1215.
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Chapter 2: Problem 16 Discrete Mathematics with Applications 4
Determine whether the statement forms in 1624 are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence.
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Chapter 2: Problem 17 Discrete Mathematics with Applications 4
Determine whether the statement forms in 1624 are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence.
Read more -
Chapter 2: Problem 18 Discrete Mathematics with Applications 4
Determine whether the statement forms in 1624 are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence.
Read more -
Chapter 2: Problem 19 Discrete Mathematics with Applications 4
Determine whether the statement forms in 1624 are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence.
Read more -
Chapter 2: Problem 20 Discrete Mathematics with Applications 4
Determine whether the statement forms in 1624 are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence.
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Chapter 2: Problem 21 Discrete Mathematics with Applications 4
Determine whether the statement forms in 1624 are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence.
Read more -
Chapter 2: Problem 22 Discrete Mathematics with Applications 4
Determine whether the statement forms in 1624 are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence.
Read more -
Chapter 2: Problem 23 Discrete Mathematics with Applications 4
Determine whether the statement forms in 1624 are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence.
Read more -
Chapter 2: Problem 24 Discrete Mathematics with Applications 4
Determine whether the statement forms in 1624 are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence.
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Chapter 2: Problem 25 Discrete Mathematics with Applications 4
Use De Morgans laws to write negations for the statements in 2531
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Chapter 2: Problem 26 Discrete Mathematics with Applications 4
Use De Morgans laws to write negations for the statements in 2531
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Chapter 2: Problem 27 Discrete Mathematics with Applications 4
Use De Morgans laws to write negations for the statements in 2531
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Chapter 2: Problem 28 Discrete Mathematics with Applications 4
Use De Morgans laws to write negations for the statements in 2531
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Chapter 2: Problem 29 Discrete Mathematics with Applications 4
Use De Morgans laws to write negations for the statements in 2531
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Chapter 2: Problem 30 Discrete Mathematics with Applications 4
Use De Morgans laws to write negations for the statements in 2531
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Chapter 2: Problem 31 Discrete Mathematics with Applications 4
Use De Morgans laws to write negations for the statements in 2531
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Chapter 2: Problem 32 Discrete Mathematics with Applications 4
Assume x is a particular real number and use De Morgans laws to write negations for the statements in 3237.
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Chapter 2: Problem 33 Discrete Mathematics with Applications 4
Assume x is a particular real number and use De Morgans laws to write negations for the statements in 3237.
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Chapter 2: Problem 34 Discrete Mathematics with Applications 4
Assume x is a particular real number and use De Morgans laws to write negations for the statements in 3237.
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Chapter 2: Problem 35 Discrete Mathematics with Applications 4
Assume x is a particular real number and use De Morgans laws to write negations for the statements in 3237.
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Chapter 2: Problem 36 Discrete Mathematics with Applications 4
Assume x is a particular real number and use De Morgans laws to write negations for the statements in 3237.
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Chapter 2: Problem 37 Discrete Mathematics with Applications 4
Assume x is a particular real number and use De Morgans laws to write negations for the statements in 3237.
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Chapter 2: Problem 38 Discrete Mathematics with Applications 4
In 38 and 39, imagine that num_orders and num_instock are particular values, such as might occur during execution of a computer program. Write negations for the following statements.
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Chapter 2: Problem 39 Discrete Mathematics with Applications 4
In 38 and 39, imagine that num_orders and num_instock are particular values, such as might occur during execution of a computer program. Write negations for the following statements.
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Chapter 2: Problem 40 Discrete Mathematics with Applications 4
Use truth tables to establish which of the statement forms in 4043 are tautologies and which are contradictions.
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Chapter 2: Problem 41 Discrete Mathematics with Applications 4
Use truth tables to establish which of the statement forms in 4043 are tautologies and which are contradictions.
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Chapter 2: Problem 42 Discrete Mathematics with Applications 4
Use truth tables to establish which of the statement forms in 4043 are tautologies and which are contradictions.
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Chapter 2: Problem 43 Discrete Mathematics with Applications 4
Use truth tables to establish which of the statement forms in 4043 are tautologies and which are contradictions.
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Chapter 2: Problem 44 Discrete Mathematics with Applications 4
In 44 and 45, determine whether the statements in (a) and (b) are logically equivalent.
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Chapter 2: Problem 45 Discrete Mathematics with Applications 4
In 44 and 45, determine whether the statements in (a) and (b) are logically equivalent.
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Chapter 2: Problem 46 Discrete Mathematics with Applications 4
In Example 2.1.4, the symbol was introduced to denote exclusive or, so p q (p q) (p q). Hence the truth table for exclusive or is as follows: p q p q T T F T F T F T T F F F a. Find simpler statement forms that are logically equivalent to p p and (p p) p. b. Is (p q) r p (q r)? Justify your answer. c. Is (p q) r (p r) (q r)? Justify your answer
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Chapter 2: Problem 47 Discrete Mathematics with Applications 4
In logic and in standard English, a double negative is equivalent to a positive. There is one fairly common English usage in which a double positive is equivalent to a negative. What is it? Can you think of others?
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Chapter 2: Problem 48 Discrete Mathematics with Applications 4
In 48 and 49 below, a logical equivalence is derived from Theorem 2.1.1. Supply a reason for each step.
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Chapter 2: Problem 49 Discrete Mathematics with Applications 4
In 48 and 49 below, a logical equivalence is derived from Theorem 2.1.1. Supply a reason for each step.
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Chapter 2: Problem 50 Discrete Mathematics with Applications 4
Use Theorem 2.1.1 to verify the logical equivalences in 5054. Supply a reason for each step.
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Chapter 2: Problem 51 Discrete Mathematics with Applications 4
Use Theorem 2.1.1 to verify the logical equivalences in 5054. Supply a reason for each step.
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Chapter 2: Problem 52 Discrete Mathematics with Applications 4
Use Theorem 2.1.1 to verify the logical equivalences in 5054. Supply a reason for each step.
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Chapter 2: Problem 53 Discrete Mathematics with Applications 4
Use Theorem 2.1.1 to verify the logical equivalences in 5054. Supply a reason for each step.
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Chapter 2: Problem 54 Discrete Mathematics with Applications 4
Use Theorem 2.1.1 to verify the logical equivalences in 5054. Supply a reason for each step.
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