Answer to Question #324393 in Discrete Mathematics for Alif

Question #324393

Prove that a real number π‘₯ is irrational if and only if 5π‘₯ is irrational


1
Expert's answer
2022-04-06T17:22:35-0400

Let  xβˆ‰Q.Suppose  5x∈Q,then  x=5x5∈Q,which  is  a  contradictory.Let  5xβˆ‰Q.Suppose  x∈Q,then  5x=5β‹…x∈Q,which  is  a  contradictory.Thusxβˆ‰Q⇔5xβˆ‰QLet\,\,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}


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