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Quantifiers and Negation

Quantifiers and NegationFor all of you, there exists information about quantifiers often quantify a variable for a statement, or predicate, by claiming a statement holdsfor allvalues of thequantity or we saythere existsa quantity for which the statement holds (at least one). Notationally, we canwrite this in shorthand as follows: x A, P(x),which claims: for allxin the setA, the statementP(x) is true. x A, P(x),which claims: there exists at least onexin the setAsuch that the statementP(x) is are many equivalent way to express these quantifiers inEnglish. Here are a few examples:Universal Quantifier: Here are a few ways to say x N: For all natural numbersx.

This statement says that the following in this exact order: 1. AT LEAST ONE y can be found BEFORE any other variable is set. And this one special y will work ... (Note: This is a false statement.) Observe: By looking at the two examples above, you should note that ∀x ∈ A,∃y ∈ B,... and ∃y ∈ B,∀x ∈ ...

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