GATE CSE 2009 | Question: 23

Question

Which one of the following is the most appropriate logical formula to represent the statement?

"Gold and silver ornaments are precious".

The following notations are used:        

• $G(x): x$ is a gold ornament
• $S(x): x$ is a silver ornament        
• $P(x): x$ is precious

• A. $\forall x(P(x) \implies (G(x) \wedge S(x)))$
• B. $\forall x((G(x) \wedge S(x)) \implies P(x))$
• C. $\exists x((G(x) \wedge S(x)) \implies P(x))$
• D. $\forall x((G(x) \vee S(x)) \implies P(x))$

Comments

@Deepak Poonia

Assume, 

p: gold ornament is precious. 

q: silver ornament is precious. 

In case of "$\vee$" p & q both can be True, i.e. we can say that p & q both are precious. 

But in case of "$\oplus$" p & q both can't be True. So, we can't say that p & q both are precious. 

That's why we can't use Exclusive-or becoz in question given statement is "Gold & Silver ornaments are precious. "

Sir, am i correct???? 

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Please read & analyze what I have written in the previous comments. You are asking your pre-determined doubts without analyzing what I have written. 

1.  Watch This Lecture till time 00:33:20.

2.  After watching the above lecture, read This Comment.

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Assume the domain contains boys and girls. In the domain, an element is either a boy or girl but not both. 

Now, the statement “everyone is either a boy or a girl” can be expressed as: 

$\forall x (B(x) \oplus G(x)) $ Or $ \forall x (B(x) \vee G(x))$.. Both are correct. 

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Also read ​​​​​ This Comment.

Similar question: GATE CSE 2006 | Question: 26

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@Deepak Poonia Thanks Sir..... 

Actually I didn't know this concept. Thanks again..... 

Answer 1

By default when no quantifier words is present then it is assumed that it is “All”

Option D is correct.

Answer 2

Option D

First thing that we need to keep in mind is to decide the domain. Since, they did'nt specified what x is,  we will take x as all ornaments in the universe.

Second thing, Here in the question the word and is misleading & it does'nt mean that the ornament will be Gold & Silver. It can either be Gold or Silver.

Third is correct choice of quantifier.

if we use existential quantifier, it will mean that at least one Gold or Silver ornament is precious, which is not what the statement is saying. if the ornament is Gold or Silver then it will be precious.(all Gold or Silver ornament are precious). So we have to use universal quantifier.

 

Answer 3

 

Answer(D)

∀x((G(x)∨S(x))⟹P(x))

 

we used ∀x because all metals which are gold or silver are precious

∨ is used as metal can either be Gold or silver or other we cannot use AND because no metal can be both gold and silver

⟹ is used as we don’t care for other elements if an elements is not gold or silver then implication will give true result and ∀x to be everything should be true for gold or silver 

Answer 4

 

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