Proving Equivalence of Rational Number Statements

  • Context: Undergrad 
  • Thread starter Thread starter ArtyJay9
  • Start date Start date
  • Tags Tags
    Proof
Click For Summary
SUMMARY

The discussion establishes the equivalence of three statements regarding rational numbers: (i) x is rational, (ii) x/2 is rational, and (iii) 3x - 1 is rational. It concludes that if x can be expressed as p/q (where p and q are integers and q is non-zero), then x/2 can be represented as m/n for appropriate integers m and n. The method of proof involves demonstrating that each statement implies the next, thereby confirming their equivalence through logical deduction.

PREREQUISITES
  • Understanding of rational numbers and their properties
  • Familiarity with mathematical proof techniques, particularly proof by contradiction
  • Knowledge of integer representation and operations
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study proof by contradiction in mathematical logic
  • Explore properties of rational numbers in depth
  • Learn about equivalence relations in mathematics
  • Investigate the implications of rational number operations
USEFUL FOR

Mathematics students, educators, and anyone interested in number theory or logical proofs related to rational numbers.

ArtyJay9
Messages
1
Reaction score
0
Show that these statement about real number x are equivalent
(i) x is rational
(ii) x/2 is rational
(iii)3x-1is rational

i think you have to use contradiction but i don't know how
 
Physics news on Phys.org
x is rational if and only if you can write it in the form p/q for p, q integers (and q non-zero). So if x is rational, let x=p/q. Can you find integers m and n such that x/2=m/n? You can do this exact process to show that 1 implies 2, 2 implies 3, and 3 implies 1 (which gives that they are equivalent)
 

Similar threads

Undergrad I don't understand Dedekind cuts
  • · Replies 14 ·
Replies
14
Views
1K
High School Uncomputable Numbers and Rational Approximations
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Proof of a fact about real numbers - transcendental and algebraic
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Error(?) in proof that the rational numbers are denumerable
  • · Replies 3 ·
Replies
3
Views
2K
High School Another consequence of Cantor's diagonal argument
  • · Replies 43 ·
2
Replies
43
Views
6K
High School Proof That ##\sqrt{x}## Isn't Rational (Unless ##x## is a Perfect Square)
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Quantifiers with integers and rational numbers
  • · Replies 2 ·
Replies
2
Views
2K
MHB Proving Countability: Positive Rational Numbers & Union of Countable Sets
  • · Replies 11 ·
Replies
11
Views
3K
MHB Is (27/4)/(6.75) a Whole, Natural, Integer, Rational, or Irrational Number?
  • · Replies 4 ·
Replies
4
Views
3K
MHB Proving Rationality of Squares of 9th Day 2 ARO 2004/2005 Numbers
  • · Replies 5 ·
Replies
5
Views
2K