- #1
Aziza
- 190
- 1
What is the difference between the following two questions:
(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?
(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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