GATE IT 2004 | Question: 3

Question

Let $a(x, y), b(x, y,)$ and $c(x, y)$ be three statements with variables $x$ and $y$ chosen from some universe. Consider the following statement:

$\qquad(\exists x)(\forall y)[(a(x, y) \wedge b(x, y)) \wedge \neg c(x, y)]$

Which one of the following is its equivalent?

• A. $(\forall x)(\exists y)[(a(x, y) \vee b(x, y)) \to c(x, y)]$
• B. $(\exists x)(\forall y)[(a(x, y) \vee b(x, y)) \wedge\neg c(x, y)]$
• C. $\neg (\forall x)(\exists y)[(a(x, y) \wedge b(x, y)) \to c(x, y)]$
• D. $\neg (\forall x)(\exists y)[(a(x, y) \vee b(x, y)) \to c(x, y)]$

Answer 1

$(\exists x)(\forall y)[(a(x, y) \wedge b(x, y)) \wedge ¬c(x, y)]$

$\quad \equiv ¬(\forall x)¬(\forall y)[(a(x, y) \wedge b(x, y)) \wedge ¬c(x, y)]$
$\quad (\because (\exists x) F(x) = \neg \forall x \neg F(x))$

$\quad \equiv ¬(\forall x)(\exists y)¬[(a(x, y) \wedge b(x, y)) \wedge ¬c(x, y)]$
$\quad (\because (\forall x) F(x) = \neg \exists x \neg F(x)), \neg \neg F(x) = F(x))$

$\quad \equiv ¬(\forall x)(\exists y)[¬(a(x, y) \wedge b(x, y)) \vee c(x, y)]$

$\quad \equiv ¬(\forall x)(\exists y)[(a(x, y) \wedge b(x, y)) → c(x, y)]$

(C) choice.

Comments

srestha

what is now not in syllabus??

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@akash

I thought it is logic programming

like this https://gateoverflow.in/958/gate2003-71

sorry misunderstood

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@Arjun sir, do we have to check the logical equivalence for each option like this? because, then it will consume quite a good amount of time. Any other quick way to figure it out?

Answer 2

.