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Consider the following well-formed formulae:
 
Consider the following well-formed formulae:
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I. $&not;&forall;x(P(x))$
 
I. $&not;&forall;x(P(x))$
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II. $&not;&exist;x(P(x))$
 
II. $&not;&exist;x(P(x))$
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III. $&not;&exist;x(&not;P(x))$
 
III. $&not;&exist;x(&not;P(x))$
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IV. $&exist;x(&not;P(x))$
 
IV. $&exist;x(&not;P(x))$
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Which of the above are equivalent?
 
Which of the above are equivalent?
 
 
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==={{Template:Author|Happy Mittal|{{mittalweb}} }}===
 
==={{Template:Author|Happy Mittal|{{mittalweb}} }}===
 
A formula $&forall;x(P(x))$ is equivalent to formula $&not;&exist;x(&not;P(x))$ i.e. add $&not;$ inside and outside, and
 
A formula $&forall;x(P(x))$ is equivalent to formula $&not;&exist;x(&not;P(x))$ i.e. add $&not;$ inside and outside, and
convert $&forall;$ to &exist;$.
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convert $&forall;$ to $&exist;$.
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So, $&not;&forall;x(P(x))$ is equivalent to $&exist;x(&not;P(x))$.  
 
So, $&not;&forall;x(P(x))$ is equivalent to $&exist;x(&not;P(x))$.  
  

Revision as of 20:03, 14 July 2014

Consider the following well-formed formulae:

I. $¬∀x(P(x))$

II. $¬∃x(P(x))$

III. $¬∃x(¬P(x))$

IV. $∃x(¬P(x))$

Which of the above are equivalent?

(A) I and III

(B) I and IV

(C) II and III

(D) II and IV

Solution by Happy Mittal

A formula $∀x(P(x))$ is equivalent to formula $¬∃x(¬P(x))$ i.e. add $¬$ inside and outside, and convert $∀$ to $∃$.

So, $¬∀x(P(x))$ is equivalent to $∃x(¬P(x))$.




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Consider the following well-formed formulae:

I. $¬∀x(P(x))$

II. $¬∃x(P(x))$

III. $¬∃x(¬P(x))$

IV. $∃x(¬P(x))$

Which of the above are equivalent?

(A) I and III

(B) I and IV

(C) II and III

(D) II and IV

Solution by Happy Mittal[edit]

A formula $∀x(P(x))$ is equivalent to formula $¬∃x(¬P(x))$ i.e. add $¬$ inside and outside, and convert $∀$ to $∃$.

So, $¬∀x(P(x))$ is equivalent to $∃x(¬P(x))$.




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