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
"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.
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."
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.
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.
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.