Question

In: Advanced Math

Let X be Z or Q and define a logical formula p by ∀x ∈ X,...

Let X be Z or Q and define a logical formula p by ∀x ∈ X, ∃y ∈ X, (x < y ∧ [∀z ∈ X, ¬(x < z ∧ z < y)]).

Describe what p asserts about the set X. Find the maximally negated logical formula equivalent to ¬p. Prove that p is true when X = Z and false when X = Q

Expert Solution

Given statement , .

Now ,

That is does not lies between and .

So asserts , " For any number in there is a number in bigger than it and there is no number between them " .

While we negate an statement is been always replaced by and is beenreplaced by that is negation of is and negation of is .

Hence the negation of the given statement is ,

If .

For each integer if we choose then and therre is no integer between and as there is no integer between and .

So the statement is true for

If

Suppose then for any choosen such that . there will a rational number between and as between any two rational number there is an rational number . Explicitly we can choose then the condition will not be satisfies and so the statement is not true for .

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If you have doubt or need more clarification at any step please comment .

Colby Messinger answered 10 months ago

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