What is the difference between the following two questions:(adsbygoogle = window.adsbygoogle || []).push({});

(a) For every positive real number x, there is a positive real number y less

than x with the property that for all positive real numbers z, yz ≥ z.

(b) For every positive real number x, there is a positive real number y with

the property that if y < z, then for all positive real numbers z, yz ≥ z.

(b) I understand as

[itex](\forall x\inℝ\stackrel{+}{})(\exists y\inℝ\stackrel{+}{})[(y<x)\Rightarrow(\forall z\inℝ\stackrel{+}{})(yz≥z)][/itex]

I am unsure of how to understand (a) but this is my interpretation:

[itex](\forall x\inℝ\stackrel{+}{})(\exists y\inℝ\stackrel{+}{})[y<x\wedge(\forall z\inℝ\stackrel{+}{})(yz≥z)][/itex]

Other than the fact that (b) has an implication and (a) does not, I do not see any difference between (a) and (b) and they both seem false to me because if you choose x=1 and 0<y<1 and z=1, then it is not the case that yz>=z. However, according to the back of my book, it says that x=1 is a counterexample to (a), not (b). It also says that (b) is actually a true statement...please help explain?

edit: I think I see why (b) is true..is it because for all x, if you choose y>x, then y<x is false, and so false=>false and false=>true are both true ?

So then x=1 would just be a counterexample to (a). But am I expressing (a) correctly?

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# Logic: difference between very similar statements

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