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Chapter 1: Problem 6

Use a truth table to verify the first De Morgan law \(\neg(p \wedge q) \equiv \neg p \vee \neg q .\)

Short Answer

Expert verified
The truth table shows that \( \eg(p \wedge q) \) and \( \eg p \vee \eg q \) have the same truth values, verifying De Morgan's first law.

Step by step solution

01

- Set up variables

Identify the variables in the expression: Let \( p \) and \( q \) represent the two propositions.
02

- Create a truth table

Create a table with columns for all possible truth values of \( p \) and \( q \), and for the expressions \( p \wedge q \), \( \eg(p \wedge q) \), \( \eg p \), \( \eg q \), and \( \eg p \vee \eg q \).
03

- Fill in truth values for \( p \) and \( q \)

Fill in all combinations of truth values for \( p \) and \( q \): True (T) and False (F).
04

- Calculate \( p \wedge q \)

Compute the conjunction \( p \wedge q \) for each row of the truth table.
05

- Calculate \( \eg(p \wedge q) \)

Negate the result of \( p \wedge q \) for each row to get \( \eg(p \wedge q) \).
06

- Calculate \( \eg p \)

Negate the value of \( p \) for each row to get \( \eg p \).
07

- Calculate \( \eg q \)

Negate the value of \( q \) for each row to get \( \eg q \).
08

- Calculate \( \eg p \vee \eg q \)

Compute the disjunction \( \eg p \vee \eg q \) using the values from \( \eg p \) and \( \eg q \).
09

- Compare results

Compare the columns \( \eg(p \wedge q) \) and \( \eg p \vee \eg q \) to verify if they are the same for all combinations of \( p \) and \( q \). If they match in all cases, the first De Morgan's law is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

truth table
A key tool for analyzing logical expressions is the truth table. It's a simple way to map out the truth values of propositions and their combinations. Imagine you have two propositions: p and q. There are four possible combinations of truth values these propositions can take: both true (T, T), p true and q false (T, F), p false and q true (F, T), and both false (F, F). By laying out these combinations in a table, you can systematically explore how the truth values change based on logical operations applied to these propositions. For example, in this exercise with De Morgan's laws, we created a table with columns for p, q, p 鈭� q, 卢(p 鈭� q), 卢p, 卢q, and 卢p 鈭� 卢q, and filled in each column step by step.
logical equivalence
Logical equivalence is a fundamental concept in logic that means two expressions are equal in truth value under all possible circumstances. When we say that \( eg(p \wedge q) \equiv \eg p \vee \eg q \), we mean that both expressions will produce the same truth values no matter what specific truth values p and q take. Thus, our task was to verify this equivalence using a truth table. After filling out all the necessary values, we compared the columns 卢(p 鈭� q) and 卢p 鈭� 卢q. Since these columns matched for all possible combinations of p and q, we confirmed that the two expressions are indeed logically equivalent. Relating different logical expressions and proving their equivalence helps students understand how complex logical statements can be simplified.
negation
Negation is a basic operation in logic, often represented by 卢 or ~, which flips the truth value of a proposition. If p is true, then 卢p is false, and if p is false, then 卢p is true. This operation is crucial in the context of De Morgan's laws. In the exercise, we negated individual propositions (卢p and 卢q) as well as more complex expressions (卢(p 鈭� q)). For example, if p and q are both true (T, T), their conjunction p 鈭� q is true, but negating it, 卢(p 鈭� q), would turn it false. Understanding negation and how it interacts with other logical operators (like conjunction 鈭� and disjunction 鈭�) is essential for solving logical equivalences and simplifying logical statements.

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Most popular questions from this chapter

Use a proof by exhaustion to show that a tiling using dominoes of a \(4 \times 4\) checkerboard with opposite corners removed does not exist. [Hint: First show that you can assume that the squares in the upper left and lower right corners are removed. Number the squares of the original checkerboard from 1 to \(16,\) starting in the first row, moving right in this row, then starting in the leftmost square in the second row and moving right, and so on. Remove squares 1 and \(16 .\) To begin the proof, note that square 2 is covered either by a domino laid horizontally, which covers squares 2 and \(3,\) or vertically, which covers squares 2 and \(6 .\) Consider each of these cases separately, and work through all the subcases that arise. \(]\)

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Exercises \(61-64\) are based on questions found in the book Symbolic Logic by Lewis Carroll. Let P(x), Q(x), R(x), and S(x) be the statements 鈥渪 is a duck,鈥� 鈥渪 is one of my poultry,鈥� 鈥渪 is an officer,鈥� and 鈥渪 is willing to waltz,鈥� respectively. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), R(x), and S(x). a) No ducks are willing to waltz. b) No officers ever decline to waltz. c) All my poultry are ducks. d) My poultry are not officers. e) Does (d) follow from (a), (b), and (c)? If not, is there a correct conclusion?

For each of these arguments, explain which rules of inference are used for each step. a) 鈥淒oug, a student in this class, knows how to write programs in JAVA. Everyone who knows how to write programs in JAVA can get a high-paying job. Therefore, someone in this class can get a high-paying job.鈥� b) 鈥淪omebody in this class enjoys whale watching. Every person who enjoys whale watching cares about ocean pollution. Therefore, there is a person in this class who cares about ocean pollution.鈥� c) 鈥淓ach of the 93 students in this class owns a personal computer. Everyone who owns a personal computer can use a word processing program. Therefore, Zeke, a student in this class, can use a word processing pro- gram.鈥� d) 鈥淓veryone in New Jersey lives within 50 miles of the ocean. Someone in New Jersey has never seen the ocean. Therefore, someone who lives within 50 miles of the ocean has never seen the ocean.鈥�

Express each of these statements using predicates and quantifiers. a) A passenger on an airline qualifies as an elite flyer if the passenger flies more than 25,000 miles in a year or takes more than 25 flights during that year. b) A man qualifies for the marathon if his best previous time is less than 3 hours and a woman qualifies for the marathon if her best previous time is less than 3.5 hours. c) A student must take at least 60 course hours, or at least 45 course hours and write a master鈥檚 thesis, and receive a grade no lower than a B in all required courses, to receive a master鈥檚 degree. d) There is a student who has taken more than 21 credit hours in a semester and received all A鈥檚.
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