Prove that a real number 𝑥 is irrational if and only if 5𝑥 is irrational
Let x∉Q.Suppose 5x∈Q,then x=5x5∈Q,which is a contradictory.Let 5x∉Q.Suppose x∈Q,then 5x=5⋅x∈Q,which is a contradictory.Thusx∉Q⇔5x∉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}Letx∈/Q.Suppose5x∈Q,thenx=55x∈Q,whichisacontradictory.Let5x∈/Q.Supposex∈Q,then5x=5⋅x∈Q,whichisacontradictory.Thusx∈/Q⇔5x∈/Q
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