GO Classes CS Test Series 2025 | Discrete Mathematics | Topic Wise Test 1 | Question: 3

Answer 1

“There is a successful person who is grateful”

Suppose, S(x) = “x is Successful” and G(x) = “x is Grateful”

It can be written as: $\exists x \left ( S\left ( x \right ) \wedge G\left ( x \right )\right )$

It’s negation would be,

= $\sim \left [ \exists x \left ( S\left ( x \right ) \wedge G\left ( x \right )\right )\right ]$

= $\forall x\left ( \sim S\left ( x \right ) \vee \sim G\left ( x \right )\right )$

= $\forall x\left ( S\left ( x \right )\rightarrow \sim G\left ( x \right ) \right )$

= “Every successful person in ungreatful”

 

$\forall x\left ( \sim S\left ( x \right ) \vee \sim G\left ( x \right )\right )$ can also be written as,

= $\forall x\left ( G\left ( x \right )\rightarrow \sim S\left ( x \right ) \right )$

= “Every greatful person in unsuccessful”

 

$Answer: \left ( B \right )   and \left ( D \right )$

Comments

B,D are correct but I marked A as well since if “EVERY successful person is ungrateful”,then “ there is a successful person who is ungrateful “ is definitely TRUE

Answer 2

If “Every successful person is ungrateful”

 

then option A should also be correct.

 

“There is a successful person who is ungrateful."

Comments

@Deepak Poonia Sir, but it is not given in the question about the empty set possibility. So then should we consider option A?

Shouldnt this rule work here too – $\forall x P(x) \rightarrow \exists x P(x)$

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@Deepak Poonia I think I have mistaken the A as : $\exists x(S(x) \rightarrow \neg G(x))$

Why is A: $\exists x(S(x) \wedge \neg G(x))$  ??

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@rigved25, Option A is $\exists x [ S(x) \wedge \neg G(x) ]$. Watch HERE.