Let S be the statement "The square of any rational number is rational." A formal version of S is "For every rational number r, r² is rational." Fill in the blanks in the proof for S. Proof: Suppose that r is (a) By definition of rational, r = a/b for some (b) with b 0. By = substitution, p² (c) =a²/b². = Since a and b are both integers, so are the prod- ucts a² and_ (d). Also b² #0 by the (e) Hence r² is a ratio of two integers with a non- zero denominator, and so rational. by definition of
Let S be the statement "The square of any rational number is rational." A formal version of S is "For every rational number r, r² is rational." Fill in the blanks in the proof for S. Proof: Suppose that r is (a) By definition of rational, r = a/b for some (b) with b 0. By = substitution, p² (c) =a²/b². = Since a and b are both integers, so are the prod- ucts a² and_ (d). Also b² #0 by the (e) Hence r² is a ratio of two integers with a non- zero denominator, and so rational. by definition of
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
#12 please
![**Exercise 11**
Prove that every integer is a rational number.
**Exercise 12**
Let \( S \) be the statement "The square of any rational number is rational." A formal version of \( S \) is "For every rational number \( r \), \( r^2 \) is rational." Fill in the blanks in the proof for \( S \).
**Proof:**
Suppose that \( r \) is \((a)\). By definition of rational, \( r = a/b \) for some \((b)\) with \( b \neq 0 \). By substitution,
\[ r^2 = \frac{(c)}{} = \frac{a^2}{b^2}. \]
Since \( a \) and \( b \) are both integers, so are the products \( a^2 \) and \((d)\). Also \( b^2 \neq 0 \) by the \((e)\). Hence \( r^2 \) is a ratio of two integers with a non-zero denominator, and so \((f)\) by definition of rational.
**Exercise 13**
Consider the following statement: The negative of any rational number is rational.
a. Write the statement formally using a quantifier and a variable.
b. Determine whether the statement is true or false and justify your answer.
**Exercise 14**
Consider the statement: The cube of any rational number is a rational number.
a. Write the statement formally using a quantifier and a variable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ec5dee8-4b81-4c1e-b6ca-023c2626018b%2F93b4c48e-0505-4192-8f2f-4cf7d330f6e7%2F5yfjhs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 11**
Prove that every integer is a rational number.
**Exercise 12**
Let \( S \) be the statement "The square of any rational number is rational." A formal version of \( S \) is "For every rational number \( r \), \( r^2 \) is rational." Fill in the blanks in the proof for \( S \).
**Proof:**
Suppose that \( r \) is \((a)\). By definition of rational, \( r = a/b \) for some \((b)\) with \( b \neq 0 \). By substitution,
\[ r^2 = \frac{(c)}{} = \frac{a^2}{b^2}. \]
Since \( a \) and \( b \) are both integers, so are the products \( a^2 \) and \((d)\). Also \( b^2 \neq 0 \) by the \((e)\). Hence \( r^2 \) is a ratio of two integers with a non-zero denominator, and so \((f)\) by definition of rational.
**Exercise 13**
Consider the following statement: The negative of any rational number is rational.
a. Write the statement formally using a quantifier and a variable.
b. Determine whether the statement is true or false and justify your answer.
**Exercise 14**
Consider the statement: The cube of any rational number is a rational number.
a. Write the statement formally using a quantifier and a variable.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
