### Logic and Set Theory: Analyzing Statements**Problem Statement:**Given that $ Q(x) $ is the statement $ x + 1 > 2x $, and the domain is the set of positive integers, which of the following best characterizes the two statements below?A) There exists $ x $ such that $ Q(x) $ is true.B) For every $ x $, $ \neg Q(x) $ is true.**Options:**1. Only A is true.2. Only B is true.3. Both A and B are true.4. Both A and B are false.**Analysis:**Firstly, we analyze the given statement $ Q(x) $:\[Q(x): x + 1 > 2x\]Simplifying $ Q(x) $:\[x + 1 > 2x \implies 1 > x\]Since $ x $ is a positive integer (from the domain definition), $ x $ must be greater than or equal to 1. Therefore, the inequality $ 1 > x $ is false for all positive integers $ x $.**Investigating Statement A:**Statement A posits that there exists an $ x $ such that $ Q(x) $ is true:\[\exists x (x + 1 > 2x)\]Because $ Q(x) $ simplifies to $ 1 > x $, and there is no positive integer $ x $ for which this is true, Statement A is false.**Investigating Statement B:**Statement B posits that for every $ x $, the negation of $ Q(x) $ is true:\[\forall x (\neg Q(x))\]Since $ Q(x) $ is equivalent to $ 1 > x $, its negation is:\[\neg Q(x): x \geq 1\]This inequality $ x \geq 1 $ is true for all positive integers $ x $, hence Statement B is true.**Conclusion:**From the analysis, we can conclude that "Only B is true."Thus, the correct option is:\[\boxed{\text{Only B is true.}}\]

If Q(x) is the statement x + 1 > 2x, and the domain is the set of positive integers, which of the following best characterizes the following two statements : A) there exists x such that Q(x) is true B) for every x, -Q(x) is true Only A is true Only B is true Both A and B are true Both A and B are false

If Q(x) is the statement x + 1 > 2x, and the domain is the set of positive integers, which of the following best characterizes the following two statements : A) there exists x such that Q(x) is true B) for every x, -Q(x) is true Only A is true Only B is true Both A and B are true Both A and B are false

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Concept explainers

Bare bones programming language

Bare bones programming language is a low-level programming language that gives a detailed stepwise description of a program. It has a tight program design and in-depth understanding of the toolchain. It is based on the target environment and is used for safe coding practices. C programming language is a type of bare bones programming language.

Question

### Logic and Set Theory: Analyzing Statements

**Problem Statement:**
Given that $ Q(x) $ is the statement $ x + 1 > 2x $, and the domain is the set of positive integers, which of the following best characterizes the two statements below?

A) There exists $ x $ such that $ Q(x) $ is true.

B) For every $ x $, $ \neg Q(x) $ is true.

**Options:**
1. Only A is true.
2. Only B is true.
3. Both A and B are true.
4. Both A and B are false.

**Analysis:**

Firstly, we analyze the given statement $ Q(x) $:
\[
Q(x): x + 1 > 2x
\]

Simplifying $ Q(x) $:
\[
x + 1 > 2x \implies 1 > x
\]

Since $ x $ is a positive integer (from the domain definition), $ x $ must be greater than or equal to 1. Therefore, the inequality $ 1 > x $ is false for all positive integers $ x $.

**Investigating Statement A:**
Statement A posits that there exists an $ x $ such that $ Q(x) $ is true:
\[
\exists x (x + 1 > 2x)
\]

Because $ Q(x) $ simplifies to $ 1 > x $, and there is no positive integer $ x $ for which this is true, Statement A is false.

**Investigating Statement B:**
Statement B posits that for every $ x $, the negation of $ Q(x) $ is true:
\[
\forall x (\neg Q(x))
\]

Since $ Q(x) $ is equivalent to $ 1 > x $, its negation is:
\[
\neg Q(x): x \geq 1
\]

This inequality $ x \geq 1 $ is true for all positive integers $ x $, hence Statement B is true.

**Conclusion:**
From the analysis, we can conclude that "Only B is true."

Thus, the correct option is:
\[
\boxed{\text{Only B is true.}}
\]

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