MHB Partial Order Relation on Positive Rational Numbers and Numbers Greater Than 1/2

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The discussion focuses on the relation r defined on the set of positive rational numbers Q+, where (x,y) ∈ r if x/y is an integer. Participants are tasked with demonstrating that r is a partial order by proving reflexivity, antisymmetry, and transitivity. Additionally, they need to identify all positive rational numbers greater than 1/2 within this relation. A suggestion is made to analyze the fraction p/q, where p and q are coprime, to derive insights about their relationship when greater than 1/2. The conversation emphasizes the mathematical properties of the relation and the implications for identifying specific rational numbers.
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need help on this ..any suggestions are highly appreciatedConsider the set of positive rational numbers Q+ . Consider the relation r defined by
(x,y) ∈ r<=> x/y ∈ Z. Show that r is a partial order and determine all numbers greater than 1/2.
 
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sbrajagopal2690 said:
Consider the set of positive rational numbers Q+ . Consider the relation r defined by (x,y) ∈ r<=> x/y ∈ Z. Show that r is a partial order and determine all numbers greater than 1/2.
You must show that this relation is
a R a (reflexivity) for all;
if a R b and b R a then a = b (antisymmetry);
if a R b and R ≤ c then a R c (transitivity).

I have no idea what "determine all numbers greater than 1/2" could mean?
 
sbrajagopal2690 said:
... determine all numbers greater than 1/2.
Suppose that $p/q$ is greater than $1/2$ in this ordering (where $p/q$ is a fraction in its reduced form, so that $p$ and $q$ have no common factors other than $1$). Then $\left.\frac12\middle/\frac pq\right.$ is an integer. Simplify that compound fraction and see what that tells you about $p$ and $q$.
 
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