Prove that if a, b, and c are positive real numbers with ab = c, then a < Vc or b< Vc.

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 92E
icon
Related questions
Question
question 6。mathematical reasoning. Show steps,probably by p or q is equivalent to not p =》q
1. Prove by contradiction that 6n + 5 is odd for all integers n.
2. Prove that for all integers n, if 3n + 5 is even then n is odd. (Hint: prove the contra-
positive)
3. Prove that
|x + y| < ]x| + \y]
for all real numbers x and
Y.
4. Prove that there does not exist a smallest positive real number. (In other words, prove
that there does not exist a positive real number x such that x < y for all positive real
numbers y).
5. Recall that an irrational number is a real number which is not rational. Prove that
if x is rational and y is irrational, then x+y is irrational. You may use the fact that
the rational numbers are closed under addition - if a and b are rational numbers, then
a + b is rational as well.
6. Prove that if a, b, and c are positive real numbers with ab = c, then a <Vc or b < Vc.
7. Prove that
1
i=1
for all positive integers n.
8. Prove that
i · i! = (n+ 1)! – 1
i=1
for all positive integers n.
9. Prove that
2" < n!
for all positive integers n such that n > 4.
Transcribed Image Text:1. Prove by contradiction that 6n + 5 is odd for all integers n. 2. Prove that for all integers n, if 3n + 5 is even then n is odd. (Hint: prove the contra- positive) 3. Prove that |x + y| < ]x| + \y] for all real numbers x and Y. 4. Prove that there does not exist a smallest positive real number. (In other words, prove that there does not exist a positive real number x such that x < y for all positive real numbers y). 5. Recall that an irrational number is a real number which is not rational. Prove that if x is rational and y is irrational, then x+y is irrational. You may use the fact that the rational numbers are closed under addition - if a and b are rational numbers, then a + b is rational as well. 6. Prove that if a, b, and c are positive real numbers with ab = c, then a <Vc or b < Vc. 7. Prove that 1 i=1 for all positive integers n. 8. Prove that i · i! = (n+ 1)! – 1 i=1 for all positive integers n. 9. Prove that 2" < n! for all positive integers n such that n > 4.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elements Of Modern Algebra
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra: Structure And Method, Book 1
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
Holt Mcdougal Larson Pre-algebra: Student Edition…
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
College Algebra
College Algebra
Algebra
ISBN:
9781337282291
Author:
Ron Larson
Publisher:
Cengage Learning
College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning