My Math Forum Notation ? and ?

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 November 25th, 2012, 03:10 AM #1 Member   Joined: Nov 2009 Posts: 90 Thanks: 0 Notation ? and ? I'm trying to learn some formal language. Say we have a function y=f(x)=2x. Then I want to say as efficiently and clearly as possible that for any x in the set of real numbers, there is one and only one y, also in the set of real numbers. I don't know the correct syntax and punctuation and perhaps I even misunderstood some symbol. Can I write $\forall x \in \mathbb{R}, \exists ! y \in \mathbb{R}$ ? Thanks!
 November 25th, 2012, 06:39 AM #2 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,767 Thanks: 5 Re: Notation ? and ? To express uniqueness using this notation, you need something like the following: For every real number x, there exists a real number y such that [(y has a certain property) and (for all real z, if z has this property, z = y)]
 November 25th, 2012, 07:33 AM #3 Member   Joined: Nov 2009 Posts: 90 Thanks: 0 Re: Notation ? and ? Thank you for your reply, but I'm having trouble making sense of it. For every real number x $\forall x \in \mathbb{R}$ there exists a real number y such that $\exists y \in \mathbb{R} :$ (y has a certain property) $y= 2x$ and $\wedge$ (for all real z, if z has this property, z = y) $\forall z \in \mathbb{R}, z=2x \Rightarrow z=y$ So, putting it all together, $\forall x \in \mathbb{R}, \exists y \in \mathbb{R} : [y= 2x \wedge \forall z \in \mathbb{R}, z=2x \Rightarrow z=y]$ which to me just looks like a mess. What am I missing? I don't have to define the function with predicate logic by the way. I just want to say that given any x, there is one and only one y. Thanks!
 November 25th, 2012, 09:36 AM #4 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,767 Thanks: 5 Re: Notation ? and ? There may be other ways, but that is how I would write it (and have, in submitted exercises)!

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