/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 49 Show that \(\neg\) and \(\vee\) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 49

Show that \(\neg\) and \(\vee\) form a functionally complete collection of logical operators.

Short Answer

Expert verified
NOT (\(\eg\)) and OR (\(\vee\)) are functionally complete because they can express AND, Implication, False, and True.

Step by step solution

01

Understand the Problem

To show that \(eg\) (NOT) and \(\vee\) (OR) form a functionally complete set of logical operators, demonstrate that they can express any other logical operator.
02

Show how to express AND (\(\wedge\))

Use De Morgan鈥檚 law, which states that \(A \wedge B = \eg(\eg A \vee \eg B)\). Therefore, \(\wedge\) can be expressed using only \(\eg\) and \(\vee\).
03

Express Implication (\(\rightarrow\))

Recall the equivalence \(A \rightarrow B = \eg A \vee B\). Implication can thus be expressed using only \(\eg\) and \(\vee\).
04

Define False (\(\bot\))

Since \(A \vee \eg A\) is always true, and its negation is false, we can define \(\bot\) (False) as \(\eg(A \vee \eg A)\).
05

Define True (\(\top\))

True \(\top\) can be defined as the negation of false: \(\top = \eg(\eg(A \vee \eg A))\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

De Morgan's laws
De Morgan's laws are fundamental rules in logic that elucidate the relationships between conjunctions (AND), disjunctions (OR), and negations (NOT).The two laws are:
  • The negation of a conjunction: \(eg(A \wedge B) = eg A \vee eg B\).
  • The negation of a disjunction: \(eg(A \vee B) = eg A \wedge eg B\).
Understanding these laws is pivotal in simplifying complex logical expressions. For example, if you need to express AND using only OR and NOT, De Morgan's laws provide the formula: \A \wedge B = eg (eg A \vee eg B)\. This shows that you can use negations and disjunctions to simulate a conjunction, proving their versatility and comprehensiveness in logical operations.
Logical conjunction (AND)
Logical conjunction, denoted by \(\wedge\), is true if and only if both operands are true.

The truth table for AND is:
  • True \(\wedge\) True = True
  • True \(\wedge\) False = False
  • False \(\wedge\) True = False
  • False \(\wedge\) False = False

Using De Morgan's laws, you can represent AND in terms of NOT and OR: \(A \wedge B = eg(eg A \vee eg B)\).This shows how understanding AND is crucial for using a functionally complete set of operators, proving that NOT and OR alone can form the foundation of logical expressions.
Logical implication
Logical implication, denoted by \(\rightarrow\), expresses a relationship where the truth of one statement ensures the truth of another.

Its truth table is:
  • True \(\rightarrow\) True = True
  • True \(\rightarrow\) False = False
  • False \(\rightarrow\) True = True
  • False \(\rightarrow\) False = True

For the implication to be false, the antecedent must be true and the consequent must be false. The formula for logical implication is \(A \rightarrow B = eg A \vee B\), illustrating that it can be expressed using only NOT and OR. This is crucial in establishing that these operators can handle all logical constructions.
Truth values
In logic, truth values are used to determine the outcome of logical statements. The most common values are True (\(\top\)) and False (\(\bot\)).

To define these values using NOT () and OR (\vee):
  • False: Defined as the negation of a tautology, \(False = eg (A \vee eg A)\).
  • True: Defined as the negation of a falsehood, \(True = eg (False)\), which translates to \(True = eg (eg (A \vee eg A))\).

These definitions show that even the fundamental truth values can be derived using the discussed logical operators, solidifying their role in a functionally complete set of operations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. a) Someone in your class can speak Hindi. b) Everyone in your class is friendly. c) There is a person in your class who was not born in California. d) A student in your class has been in a movie. e) No student in your class has taken a course in logic programming.

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 baby,鈥� 鈥渪 is logical,鈥� 鈥渪 is able to manage a crocodile,鈥� and 鈥渪 is despised,鈥� respectively. Suppose that the domain consists of all people. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), R(x), and S(x). a) Babies are illogical. b) Nobody is despised who can manage a crocodile. c) Illogical persons are despised. d) Babies cannot manage crocodiles. e) Does (d) follow from (a), (b), and (c)? If not, is there a correct conclusion?

Prove that there are no solutions in positive integers \(x\) and \(y\) to the equation \(x^{4}+y^{4}=625\)

Suppose that \(a\) and \(b\) are odd integers with \(a \neq b .\) Show there is a unique integer \(c\) such that \(|a-c|=|b-c|\)

Determine whether \(\forall x(P(x) \leftrightarrow Q(x))\) and \(\forall x P(x) \leftrightarrow\) \(\forall x Q(x)\) are logically equivalent. Justify your answer.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.