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Rewrite the following statements and their negations using quantifiers and symbols. Then, argue whether the statement or its negation is true by proving the statement or by finding a counterexample. (a) There is a rational number which is greater than its square root. i. Original statement: ii. Negation: iii. Which is true and why? (b) Each integer has the property that its square is less than or equal to its cube. i. Original statement: ii. Negation: iii. Which is true and why? (c) Every element in the set {0, 1, 2} is greater than or equal to half of its square. i. Original statement: ii. Negation: iii. Which is true and why?

Rewrite the following statements and their negations using quantifiers and symbols. Then, argue whether the statement or its negation is true by proving the statement or by finding a counterexample. (a) There is a rational number which is greater than its square root. i. Original statement: ii. Negation: iii. Which is true and why? (b) Each integer has the property that its square is less than or equal to its cube. i. Original statement: ii. Negation: iii. Which is true and why? (c) Every element in the set {0, 1, 2} is greater than or equal to half of its square. i. Original statement: ii. Negation: iii. Which is true and why?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Exercise: Proving Statements and Their Negations #### Instructions: Rewrite the following statements and their negations using quantifiers and symbols. Then, argue whether the statement or its negation is true by proving the statement or by finding a counterexample. --- ### (a) Rational Number and Square Root 1. **Original statement:** - There is a rational number which is greater than its square root. 2. **Negation:** - For all rational numbers, they are not greater than their square root. 3. **Which is true and why?** - **Analysis required:** To determine the truth, examine specific rational numbers and compare them with their square roots. --- ### (b) Integer Properties of Squares and Cubes 1. **Original statement:** - Each integer has the property that its square is less than or equal to its cube. 2. **Negation:** - There exists an integer such that its square is greater than its cube. 3. **Which is true and why?** - **Analysis required:** To determine the truth, evaluate specific examples of integers and their squares and cubes to see if the relationship holds or fails. --- ### (c) Set Elements and Their Squares 1. **Original statement:** - Every element in the set {0, 1, 2} is greater than or equal to half of its square. 2. **Negation:** - There exists an element in the set {0, 1, 2} that is not greater than or equal to half of its square. 3. **Which is true and why?** - **Analysis required:** To determine the truth, compute the values of the elements {0, 1, 2} and compare them to half of their squares. --- ### Method of Analysis: To complete this exercise, one should: 1. Express each statement formally with mathematical symbols. 2. Perform calculations or logical reasoning to evaluate the truth of each statement. 3. Provide counterexamples if applicable for the negations or statements. This exercise helps to understand the application of quantifiers and negation in mathematical proofs and logical reasoning, essential for advanced mathematics and logic courses.
Transcribed Image Text:### Exercise: Proving Statements and Their Negations #### Instructions: Rewrite the following statements and their negations using quantifiers and symbols. Then, argue whether the statement or its negation is true by proving the statement or by finding a counterexample. --- ### (a) Rational Number and Square Root 1. **Original statement:** - There is a rational number which is greater than its square root. 2. **Negation:** - For all rational numbers, they are not greater than their square root. 3. **Which is true and why?** - **Analysis required:** To determine the truth, examine specific rational numbers and compare them with their square roots. --- ### (b) Integer Properties of Squares and Cubes 1. **Original statement:** - Each integer has the property that its square is less than or equal to its cube. 2. **Negation:** - There exists an integer such that its square is greater than its cube. 3. **Which is true and why?** - **Analysis required:** To determine the truth, evaluate specific examples of integers and their squares and cubes to see if the relationship holds or fails. --- ### (c) Set Elements and Their Squares 1. **Original statement:** - Every element in the set {0, 1, 2} is greater than or equal to half of its square. 2. **Negation:** - There exists an element in the set {0, 1, 2} that is not greater than or equal to half of its square. 3. **Which is true and why?** - **Analysis required:** To determine the truth, compute the values of the elements {0, 1, 2} and compare them to half of their squares. --- ### Method of Analysis: To complete this exercise, one should: 1. Express each statement formally with mathematical symbols. 2. Perform calculations or logical reasoning to evaluate the truth of each statement. 3. Provide counterexamples if applicable for the negations or statements. This exercise helps to understand the application of quantifiers and negation in mathematical proofs and logical reasoning, essential for advanced mathematics and logic courses.
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