A Logic Problem

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0
1. The problem statement, all variables and given/known data
There exists y > 0 such that [[tex]y^{2} = x[/tex] if and only if [tex]x > 0[/tex]].

This means that "there is some positive number whose square equals all positive
numbers." - St. John College, Oxford

3. The attempt at a solution
I am not sure about this statement "- - some positive number whose square equals
all positive numbers", and particularly about the word "all".

I would read the statement as
If [tex] \exists y > 0[/tex], then [tex]\exists [ y^{2} = x[/tex] if and only if [tex]x > 0] [/tex]

It seems that the statement should be read as
If [tex] \exists y > 0[/tex], then [tex]\forall [ y^{2} = x[/tex] if and only if [tex]x > 0] [/tex]

Is there always "for all" after "such that"?
 

HallsofIvy

Science Advisor
41,626
821
1. The problem statement, all variables and given/known data
There exists y > 0 such that [[tex]y^{2} = x[/tex] if and only if [tex]x > 0[/tex]].

This means that "there is some positive number whose square equals all positive
numbers." - St. John College, Oxford

3. The attempt at a solution
I am not sure about this statement "- - some positive number whose square equals
all positive numbers", and particularly about the word "all".

I would read the statement as
If [tex] \exists y > 0[/tex], then [tex]\exists [ y^{2} = x[/tex] if and only if [tex]x > 0] [/tex]
You don't say "there exists" a statement. "There exists" and "for all" only apply to variables.

It seems that the statement should be read as
If [tex] \exists y > 0[/tex], then [tex]\forall [ y^{2} = x[/tex] if and only if [tex]x > 0] [/tex]

Is there always "for all" after "such that"?
Not necessarily. There exist x> 0 such that x2= 4. That has no "for all". Try thinking about what "for all" means rather than looking for general rules.
 
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Try thinking about what "for all" means rather than looking for general rules.
It seems that we need to make statements true for a given context.

For example, the above example with "for all" is false, whereas right with the "exists". It is nonsense to say that there exists one positive real number whose square equals all positive numbers.

You don't say "there exists" a statement. "There exists" and "for all" only apply to variables.
The quantifiers apply to the variables. I agree with you about that.
 

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