in Mathematical Logic
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3 votes
3 votes

Assume the following predicate and constant symbols: 

W(x,y): x wrote y

L(x,y): x is longer than y

N(x): x is a novel

a: Amit     h: Harshal

Which of the following predicate logic formula represents the sentence:

"Harshal wrote a novel which is longer than any of the Amit's novels"

  1. $∀x∃y\left(L\left(x,y\right) → W\left(x,y\right) ∧ W\left(a,x\right)\right)$
  2. $\forall x \forall y \left(W\left(h,x\right) \wedge W\left(a,y\right) \implies L\left(x,y\right)\right)$
  3. $\exists x \forall y \left(N(x) \wedge W\left(h,x\right) \implies N\left(y\right) \wedge W\left(a,y\right) \wedge L\left(x,y\right)\right)$
  4. $\exists x  \left(N(x) \wedge W\left(h,x\right) \wedge \forall y \left( N\left(y\right) \wedge W\left(a,y\right) \implies L\left(x,y\right)\right)\right)$
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1 Answer

5 votes
5 votes
Best answer
(A) For every book if there exists a shorter book, then harshal has written the shorter one and amit the longer one.

(B) Every book written by Harshal is longer than every book written by Amit.

(C) There exists an x such that if x is a novel written by Harshal, then all novels written by Amit are shorter than x.

(D) There exists an x such that x is a novel written by Harshal and all novels written by Amit are shorter than x.

So, (D) is the answer
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8 Comments

Why not Both C and D are correct?
There is no doubt D is correct ... bt C too can be ..isnt it?
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Because C doesn't make sure that Harshal wrote a novel. (Even if x is a poetry and not a novel, C is true)
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sir , in (c) option how u can say it can be "poetry" also as ,left side of implication saying very clearly that it is novel.!!!
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That's it. Even if LHS of implication is false, the whole formula is TRUE.
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sir, i did not understand... how left side of implication is false. :(
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Suppose there is no novel at all. Even then C option is TRUE, but our given condition is not satisfied.
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(A) For every book if there exists a shorter book, then harshal has written the shorter one and amit the longer one.

There is not Harshal in option A.
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xy(L(x,y)W(h,y)W(a,x))

it should be this in quetion.

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