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
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Transcribed Image Text:n + U. Explain why (5m IZN)/ (4n) must be a
rational number.
11. Prove that every integer is a rational number.
2
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, 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,
r² =_ (c) = a²/b².
Since a and b are both integers, so are the prod-
ucts a² and_ (d)
(d). Also b² #0 by the e
Hence r² is a ratio of two integers with a non-
zero denominator, and so(f) by definition of
rational.
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.
14. Consider the statement: The cube of any rational
number is a rational number.
a. Write the statement formally using a quantifier
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