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Using De Morgan's laws, write the negation of the following statements so that the negation never appears in front of a quantifier or of an expression involving logical connectives. a) 3x3y(P(x) → Q(y)) b) 3y(3xA(x, y) V VæB(x,y))

Using De Morgan's laws, write the negation of the following statements so that the negation never appears in front of a quantifier or of an expression involving logical connectives. a) 3x3y(P(x) → Q(y)) b) 3y(3xA(x, y) V VæB(x,y))

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Negating Formulas with Nested Quantifiers De Morgan's laws allow to rewrite the negation of a disjunction or a conjunction in a logical equivalent formula where the negation appears in front of the atomic propositions and not in front of a compound proposition: -(a A b) = ¬a V ¬6 ¬(a v b) = ¬a ^ ¬b
Transcribed Image Text:Negating Formulas with Nested Quantifiers De Morgan's laws allow to rewrite the negation of a disjunction or a conjunction in a logical equivalent formula where the negation appears in front of the atomic propositions and not in front of a compound proposition: -(a A b) = ¬a V ¬6 ¬(a v b) = ¬a ^ ¬b
We have also seen how we can use so-called De Morgan's laws for quantifiers to rewrite negations of sentences with quantifiers in a logically equivalent form where the negation operator does not appear in front the quantifier: -VxP(x) = 3x¬P(x) -JxP(x) = Vx¬P(x) Using these two forms of De Morgan's laws it is possible write the negation of formulas with nested quantifiers in a logically equivalent form where the negation never appears in front of a quantifier or of an expression involving logical connectives. For example, -Væ3y(P(x, y) A Q(x,y)) = 3xVy(¬P(x,y) V ¬Q(x, y)) Note that in the second expression the negation operator only appears in front of P(x, y) and in front of Q(x, y). Using De Morgan's laws, write the negation of the following statements so that the negation never appears in front of a quantifier or of an expression involving logical connectives. a) 3x3y(P(x) → Q(y)) b) 3y(3xA(x, y) v VæB(x, y))
Transcribed Image Text:We have also seen how we can use so-called De Morgan's laws for quantifiers to rewrite negations of sentences with quantifiers in a logically equivalent form where the negation operator does not appear in front the quantifier: -VxP(x) = 3x¬P(x) -JxP(x) = Vx¬P(x) Using these two forms of De Morgan's laws it is possible write the negation of formulas with nested quantifiers in a logically equivalent form where the negation never appears in front of a quantifier or of an expression involving logical connectives. For example, -Væ3y(P(x, y) A Q(x,y)) = 3xVy(¬P(x,y) V ¬Q(x, y)) Note that in the second expression the negation operator only appears in front of P(x, y) and in front of Q(x, y). Using De Morgan's laws, write the negation of the following statements so that the negation never appears in front of a quantifier or of an expression involving logical connectives. a) 3x3y(P(x) → Q(y)) b) 3y(3xA(x, y) v VæB(x, y))
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