in Mathematical Logic edited by
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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
  1. $\forall x(P(x) \implies (G(x) \wedge S(x)))$
  2. $\forall x((G(x) \wedge S(x)) \implies P(x))$
  3. $\exists x((G(x) \wedge S(x)) \implies P(x))$
  4. $\forall x((G(x) \vee S(x)) \implies P(x))$
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2 Comments

How can we know that an ornament cannot be made of gold and silver both. We see ornaments that have both gold in some parts and silver in other parts. We cannot judge the ornaments with this given information ryt? Moreover then what will be the meaning of this " gold or silver ornaments are precious"?
2
it  will mean all ornaments which are made up of only  both  with gold and silver  are precious (may exist but does not convey the meaning of statement in question)

 gold or silver or both (inclusive or ) ornaments are precious meaning is same as of statement  in question  so same notation
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6 Answers

47 votes
 
Best answer

The statement could be translated as, if $x$ is either Gold or Silver, then it would be precious. Rather than,

If $x$ is both Gold and Silver, as an item cannot both Gold and silver at the same time.

Hence Ans is (D).

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3 Comments

well , if we think about it an item can be both gold and silver at the same time BUT here in this context an item cant be gold and silver at same time because here gold & silver items are disjoint sets ... it's something we have to understand from the question
17
Yes "and" word is confusiing and leading to wrong answer. Ornament can not be both Gold and Silver at same time.
7
These logic are more confusing. Sometimes propositions doesn't convey any real meaning if we try to solve them according to simple english sentence then we might do it wrong.

But here we have to use intuition.. /
2
7 votes

"Gold and silver ornaments are precious"

For all x, x can be either Gold or Silver ornament then the x is precious.

3 votes

Option D this is just Same as lion and tiger question https://gateoverflow.in/989/gate2006-26 

2 votes
This statement can be expressed as => For all X, x can be either gold or silver then the ornament X is precious => For all X, (G(X) v S(x)) => P(X).
0 votes

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.

 

0 votes

 

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 

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