Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.
Audit a zero exception in a quantified product statement
Problem
Evaluate the claim “For every rational \(r\) and irrational \(u\), \(\text{ru}\) is irrational.” Test \(r=0\), then state the corrected condition if the claim fails.
Big Picture
What this problem is really about
A universal claim can be disproved by one valid counterexample, so the zero case must be tested. We’ll confirm zero is rational, multiply it by an arbitrary irrational number, classify the product, and repair the theorem by adding the exact nonzero condition that removes the exception.
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