Difference between two statements

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Homework Statement


Write each of the following as an English sentence and state whether it is true or false:
a) ∀x ∈ R, ∃y ∈ R, y^3 = x.
b) ∃y ∈ R, ∀x ∈ R, y^3 = x.

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The Attempt at a Solution


I think both say every real number has at least one cubic root. Maybe the second one says, there exist a least one cubic root for every real number?
 
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  • #2
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I think that statement b) is poorly stated and that a better math statement would be:
∃y ∈ R such that ∀x ∈ R, y^3 = x.

With that change, you should be able to make clear English statements from both a) and b) and say which are true.
 
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I think that statement b) is poorly stated and that a better math statement would be:
∃y ∈ R such that ∀x ∈ R, y^3 = x.

With that change, you should be able to make clear English statements from both a) and b) and say which are true.
I think I got it.
The first one says Every real number has at least one cubic root
The second one says There is a real number that is the cubic root of every real number

So the first one is true and the second one is false.
right?
 
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  • #4
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I think I got it.
The first one says Every real number has at least one cubic root
The second one says There is a real number that is the cubic root of every real number

So the first one is true and the second one is false.
right?
Exactly. As long as that is really what the original b) statement had in mind. I think that is what the original b) intended to say.
 
  • #5
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I think that statement b) is poorly stated and that a better math statement would be:
∃y ∈ R such that ∀x ∈ R, y^3 = x.

With that change, you should be able to make clear English statements from both a) and b) and say which are true.

Statement (b) is stated according to the usual rules of logic though. The comma between the quantifiers usually are read as "such that".
 
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Statement (b) is stated according to the usual rules of logic though. The comma between the quantifiers usually are read as "such that".
I admit that formal logic was not my field but it is not normal to interpret the coma that way in pure math. Statement a) should not be read that way.

EDIT: I have to backtrack here. It appears that if the only way to make sense of consecutive quantifiers is to interpret the comma as "such that", then it can be interpreted that way. But I think it is much better and more common to use 's.t.'. The reason is that you only know if 'such that' is the correct meaning of a coma after you have read and understood the entire statement, and have ruled out 'and', and the usual English use of the coma as a separator. But then you are giving the writer the benefit of the doubt, which might not be deserved.
 
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