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Consider the following well-formed formulae: | Consider the following well-formed formulae: | ||
<br> | <br> | ||
| − | 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. $∃x(¬P(x))$ |
<br> | <br> | ||
Which of the above are equivalent? | Which of the above are equivalent? | ||
| − | + | ||
| − | + | (A) I and III | |
| − | + | ||
| − | + | '''(B) I and IV''' | |
| − | + | ||
| − | + | (C) II and III | |
| − | + | ||
| − | + | (D) II and IV | |
==={{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 ∃$. |
<br> | <br> | ||
| − | 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 $∃x(¬P(x))$.
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)). So option (B) is correct.
GATECSE