# Difference between two statements

## 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.

## 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?

Last edited:

FactChecker
Gold Member
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 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?

• FactChecker
FactChecker
Gold Member
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.

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".

FactChecker