The discussion focuses on the relation defined on the set of positive rational numbers Q+ by (x,y) ∈ r if and only if x/y ∈ Z. This relation is established as a partial order by demonstrating reflexivity, antisymmetry, and transitivity. Additionally, participants explore the implications of determining all positive rational numbers greater than 1/2, specifically through the analysis of fractions in reduced form, leading to the conclusion that for a fraction p/q to be greater than 1/2, the condition that 2 divides p must hold.
PREREQUISITESMathematicians, educators, and students studying order relations, particularly those focused on rational numbers and their properties in advanced mathematics.
You must show that this relation issbrajagopal2690 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.
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 $q$.sbrajagopal2690 said:... determine all numbers greater than 1/2.