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As part of a problem, I must show the following:

Prove that there exists an isomorphism such that $$(\{1,-1\},\cdot)\times((0,\infty),\cdot)\simeq(\mathbb{R}-\{0\},\cdot).$$ How would one actually define a map to show this? Thanks in advance for any help!

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    hint each number has a unique sign and magnitude – 2017-02-19
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    $(a,b) \mapsto ab$ is what you're looking for. – 2017-02-19
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    @Crostul How exactly would you show that this mapping is well-defined and onto? – 2017-02-19
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    @Crostul Nevermind about well-defined; I just figured that out. But can I just assume that it is onto because every element in $\mathbb{R}-\{0\}$ can be expressed as the image of an element in the domain? – 2017-02-19
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    @A.S.Hopkins That's the general meaning of "onto", so yes. – 2017-02-19

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