Prove that a real number π₯ is irrational if and only if 5π₯ is irrational
LetββxβQ.Supposeββ5xβQ,thenββx=5x5βQ,whichββisββaββcontradictory.Letββ5xβQ.SupposeββxβQ,thenββ5x=5β xβQ,whichββisββaββcontradictory.ThusxβQβ5xβQLet\,\,x\notin \mathbb{Q} . Suppose\,\,5x\in \mathbb{Q} ,then\,\,x=\frac{5x}{5}\in \mathbb{Q} , which\,\,is\,\,a\,\,contradictory.\\Let\,\,5x\notin \mathbb{Q} . Suppose\,\,x\in \mathbb{Q} , then\,\,5x=5\cdot x\in \mathbb{Q} , which\,\,is\,\,a\,\,contradictory.\\Thus\\x\notin \mathbb{Q} \Leftrightarrow 5x\notin \mathbb{Q}Letxβ/Q.Suppose5xβQ,thenx=55xββQ,whichisacontradictory.Let5xβ/Q.SupposexβQ,then5x=5β xβQ,whichisacontradictory.Thusxβ/Qβ5xβ/Q
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