Arjun Suresh (talk | contribs) |
Arjun Suresh (talk | contribs) |
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Consider the following well-formed formulae: | Consider the following well-formed formulae: | ||
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I. $¬∀x(P(x))$ | I. $¬∀x(P(x))$ | ||
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II. $¬∃x(P(x))$ | II. $¬∃x(P(x))$ | ||
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III. $¬∃x(¬P(x))$ | III. $¬∃x(¬P(x))$ | ||
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IV. $∃x(¬P(x))$ | IV. $∃x(¬P(x))$ | ||
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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 $∀x(P(x))$ is equivalent to formula $¬∃x(¬P(x))$ i.e. add $¬$ inside and outside, and | A formula $∀x(P(x))$ is equivalent to formula $¬∃x(¬P(x))$ i.e. add $¬$ inside and outside, and | ||
| − | convert $∀$ to ∃$. | + | convert $∀$ to $∃$. |
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So, $¬∀x(P(x))$ is equivalent to $∃x(¬P(x))$. | So, $¬∀x(P(x))$ is equivalent to $∃x(¬P(x))$. | ||
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
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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