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Mixed Quantifiers

1. Homework Statement
Give your own example of a statement with two different quantifiers which changes its meaning and truth value when the order of the quantifiers is exchanged.

2. Homework Equations


3. The Attempt at a Solution
(∀x∈ℝ)(∃x∈ℝ)(xy=0) is true but (∃x∈ℝ)(∀x∈ℝ)(xy=0) is false.

Is this correct?
 

fresh_42

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1. Homework Statement
Give your own example of a statement with two different quantifiers which changes its meaning and truth value when the order of the quantifiers is exchanged.

2. Homework Equations


3. The Attempt at a Solution
(∀x∈ℝ)(∃x∈ℝ)(xy=0) is true but (∃x∈ℝ)(∀x∈ℝ)(xy=0) is false.

Is this correct?
No. First of all, you have used the same variable ##x## in both quantifiers, and haven't quantified ##y## at all. Then you used a symmetric statement, ##xy=0##. How should it depend on the ordering? ##x=0##, resp. ##y=0## makes both statements true if chosen in the existence clause.
 
No. First of all, you have used the same variable ##x## in both quantifiers, and haven't quantified ##y## at all. Then you used a symmetric statement, ##xy=0##. How should it depend on the ordering? ##x=0##, resp. ##y=0## makes both statements true if chosen in the existence clause.
My apologies for using the same variable in both quantifiers.

Would this one make sense (∀x∈ℝ)(∃y∈ℝ)(x2+y=0) is true but (∃y∈ℝ)(∀x∈ℝ)(x2+y=0)?
 

fresh_42

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My apologies for using the same variable in both quantifiers.

Would this one make sense (∀x∈ℝ)(∃y∈ℝ)(x2+y=0) is true but (∃y∈ℝ)(∀x∈ℝ)(x2+y=0)?
This works.
I.) For all ##x## is a ##y##, namely ##y=-x^2## such that ... is true.
II.) There is a real number ##y## such that for all ##x## .... is false: e.g. ##y + 0^2 \neq y + 1^2## no matter how ##y## is chosen.

Do you see the difference between the two statements? There is a neglect in the notation of the first statement. Do you see it? This isn't directed towards you, since the majority of people neglect this, too. But what would be a better statement I.)?
 

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