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Chapter 1: Problem 47

Show that the two statements \(\neg \exists x \forall y P(x, y)\) and \(\forall x \exists y \neg P(x, y),\) where both quantifiers over the first variable in \(P(x, y)\) have the same domain, and both quantifiers over the second variable in \(P(x, y)\) have the same domain, are logically equivalent.

Short Answer

Expert verified
Both statements \(eg \exists x \forall y P(x, y)\) and \(\forall x \exists y eg P(x, y)\) simplify to the same form and are logically equivalent.

Step by step solution

01

- Understand the statements

The first statement is \(eg \exists x \forall y P(x, y)\), which translates to 'It is not true that there exists an \(x\) such that for all \(y\), \(P(x, y)\) is true.' The second statement is \(\forall x \exists y eg P(x, y)\), which means 'For every \(x\), there exists a \(y\) such that \(P(x, y)\) is not true.'
02

- Apply logical negation

Apply the negation inside the first statement \(eg \exists x \forall y P(x, y)\) using De Morgan's laws. The result is \(\forall x eg \forall y P(x, y)\).
03

- Simplify using De Morgan's laws

Simplify \(eg \forall y P(x, y)\) in the previous step, which is logically equivalent to \(\exists y eg P(x, y)\). Thus, the initial statement converts to \(\forall x \exists y eg P(x, y)\).
04

- Compare the simplified statements

After simplification, both statements are \(\forall x \exists y eg P(x, y)\). Therefore, they are logically equivalent.

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Key Concepts

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

Quantifiers
Quantifiers are symbols used in logic to express the quantity of specimens in the domain of discourse that satisfy a given condition. The most common quantifiers are the existential quantifier \(\exists\) and the universal quantifier \(\forall\).

\(\exists x P(x)\) means 'there exists an \) x\) such that \ P(x)\ is true.' It asserts that some specimens within the domain satisfy the predicate \ P(x).\

On the other hand, \(\forall x P(x)\) means 'for all \ x\ within the domain, the predicate \ P(x)\ is true.' It asserts that every specimen in the domain satisfies the predicate.

Understanding how to manipulate these quantifiers is essential for logic and mathematics. For example, negating quantifiers can often be tricky but is made clear using De Morgan's laws, which state \(
  • \(\eg \exists x P(x) \equiv \forall x \eg P(x) \)
  • \(\ wherein we switch the existential quantifier for the universal one (and vice versa) and negate the predicate.
De Morgan's laws
De Morgan's laws are crucial in both set theory and logic. These laws provide a way to distribute negations across conjunctions and disjunctions or, in this case, quantifiers.

In the context of quantifiers, De Morgan's laws are applied as:

\( \eg \forall x P(x) \equiv \exists x \eg P(x) \)
\( \eg \exists x P(x) \equiv \forall x \eg P(x) \)

Let's apply these laws to understand the logical equivalence in the given exercise:
\( \eg \exists x \forall y P(x, y) \equiv \forall x \eg \forall y P(x, y) \equiv \forall x \exists y \eg P(x, y)\)

By applying De Morgan's laws step-by-step, we ultimately demonstrate the logical equivalence of the given statements through simplification.
Predicate Logic
Predicate logic, or first-order logic, extends propositional logic by including quantifiers and predicates. A predicate is a function that returns a truth value and involves variables. For example, \(P(x, y)\) might denote 'x is greater than y'.

In predicate logic:
  • Quantifiers allow us to express sentences involving 'all' or 'some' elements within a given domain.
  • Predicates offer a way to talk about the properties of objects and their relationships within a formal system.

To show logical equivalences involving predicates, you often use De Morgan's laws, negations, and logical manipulation of quantifiers. Simplifying logical statements helps in understanding their fundamental properties.

In the given problem, we directly see how predicate logic is at work. We analyze the given statements and use predicate logic techniques to simplify and demonstrate their logical equivalence.

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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. \(]\)

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.

Use resolution to show the hypotheses "Allen is a bad boy or Hillary is a good girl" and "Allen is a good boy or David is happy鈥� imply the conclusion 鈥淗illary is a good girl or David is happy.鈥�

A statement is in prenex normal form (PNF) if and only if it is of the form $$ Q_{1} x_{1} Q_{2} x_{2} \cdots Q_{k} x_{k} P\left(x_{1}, x_{2}, \ldots, x_{k}\right) $$ where each \(Q_{i}, i=1,2, \ldots, k,\) is either the existential quantifier or the universal quantifier, and \(P\left(x_{1}, \ldots, x_{k}\right)\) is a predicate involving no quantifiers. For example, \(\exists x \forall y(P(x, y) \wedge Q(y))\) is in prenex normal form, whereas \(\exists x P(x) \vee \forall x Q(x)\) is not (because the quantifiers do not all occur first). Every statement formed from propositional variables, predicates, \(\mathbf{T},\) and \(\mathbf{F}\) using logical connectives and quantifiers is equivalent to a statement in prenex normal form. Exercise 51 asks for a proof of this fact. Show how to transform an arbitrary statement to a statement in prenex normal form that is equivalent to the given statement. (Note: A formal solution of this exercise requires use of structural induction, covered in Section \(5.3 . )\)

Use quantifiers to express the distributive laws of multiplication over addition for real numbers.
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