Answer Nested Quantifiers Q5: Exist X, All Y

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In summary, the conversation discusses the notation used in a physics forum and provides a reference for learning it. The conversation also provides guidance on how to use notation in solving a problem involving negation and brackets. Ultimately, the summary concludes that the correct notation for the problem is \forall z \;Q(x,y,z).
  • #1
kramer733
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Homework Statement



Question 5.

http://cg.scs.carleton.ca/~michiel/1805/assignment1.pdf

Homework Equations





The Attempt at a Solution



I arrived here:

Exist X, All Y (P(x,y) and NEGATION (Exist z: NEGATION Q(x,y,z))

I have no idea how you guys do the notation. If anybody would like to show me how to use notation here at physics forums, it'd be great. Tried googling it but failed.
 
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  • #2
very good reference for latex notation is at

tobi.oetiker.ch/lshort/lshort.pdf

for the list of mathematical symbols go to page 72 in pdf. also on this forum , there is latex help thread which can help you.

in the problem, you are doing good, you have carried [itex]\neg[/itex] till that point, why
don't you carry it forward , inside the bracket
 
  • #3
IssacNewton said:
very good reference for latex notation is at

tobi.oetiker.ch/lshort/lshort.pdf

for the list of mathematical symbols go to page 72 in pdf. also on this forum , there is latex help thread which can help you.

in the problem, you are doing good, you have carried [itex]\neg[/itex] till that point, why
don't you carry it forward , inside the bracket

I'm not sure what happens when you negate something inside a bracket. It confused me.
 
  • #4
[tex]\neg(\exists z(\neg Q(x,y,z)))[/tex]

[tex]\forall z \;\neg (\neg Q(x,y,z))[/tex]

[tex]\forall z \;Q(x,y,z)[/tex]
 

FAQ: Answer Nested Quantifiers Q5: Exist X, All Y

What is the meaning of "Exist X, All Y" in a nested quantifier statement?

"Exist X" means that there is at least one object that satisfies the condition for X. "All Y" means that all objects satisfy the condition for Y. So, the statement "Exist X, All Y" means that there is at least one object that satisfies the condition for X and all objects satisfy the condition for Y.

Can you provide an example of a statement with "Exist X, All Y" nested quantifiers?

An example of a statement with "Exist X, All Y" nested quantifiers is: "For all students in a class, there exists at least one subject that all students have passed."

How does the order of nested quantifiers affect the truth value of a statement?

The order of nested quantifiers can affect the truth value of a statement. For example, changing the order of "Exist X" and "All Y" in the statement "Exist X, All Y" can change the meaning of the statement. In general, the order of nested quantifiers can affect the scope of the statement and the objects that it applies to.

What is the difference between "Exist X, All Y" and "All Y, Exist X"?

The difference between "Exist X, All Y" and "All Y, Exist X" is the order of the nested quantifiers. In "Exist X, All Y", the "Exist X" quantifier is on the outer layer, meaning that there exists at least one object that satisfies the condition for X, and then for all those objects, the condition for Y must also be satisfied. In "All Y, Exist X", the "All Y" quantifier is on the outer layer, meaning that for all objects that satisfy the condition for Y, there exists at least one object that satisfies the condition for X. This subtle difference can change the meaning and truth value of the statement.

Are there any special rules or strategies for dealing with nested quantifiers in mathematical proofs?

Yes, there are some rules and strategies for dealing with nested quantifiers in mathematical proofs. One common strategy is to use the "if-then" approach, where you assume the "if" part (the antecedent) and show that it leads to the "then" part (the consequent). Another strategy is to use the "counterexample" approach, where you try to find specific values for the variables that make the statement false. Additionally, it is important to pay attention to the order of the nested quantifiers and to use logical reasoning to determine the appropriate direction of proof.

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