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Consider the statement

"Not all that glitters is gold”

Predicate glitters$(x)$ is true if $x$ glitters and predicate gold$(x)$ is true if $x$ is gold.  Which one of the following logical formulae represents the above statement?

1. $\forall x: glitters (x)\Rightarrow \neg gold(x)$
2. $\forall x: gold (x)\Rightarrow glitters(x)$
3. $\exists x: gold(x)\wedge \neg glitters(x)$
4. $\exists x: glitters(x)\wedge \neg gold(x)$
edited | 980 views
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option B is not true because it says that " All gold glitters " which cannot be extracted from "Not all that glitters is gold"

we can only extract from this statement is "Some that glitters is not gold  "
We don't know about the gold that all of it glitters or not

"Not all that glitters is gold”  can be expressed as :

$\neg(\forall x(glitters(x)\implies gold(x)))$

(as restriction of universal quantification is same as universal quantification of a conditional statement.)

"Not all that glitters is gold" means "some glitters are not gold" which can be expressed as

$\exists x(glitters(x)\wedge \neg gold(x))$

(as restriction of an existential quantification is same as existential quantification of a conjunction.)

So option (D) is correct.

edited by

Option D is correct .
"Not all that glitters is gold
can be expressed as :

⇒∼(∀x(glitters(x)⇒gold(x))

⇒$∃x\neg(glitters(x)⇒gold(x))$

⇒$∃x(\neg(\neg glitters(x) \vee gold(x))$

⇒$∃x(glitters(x) \wedge \neg gold(x))$

can be expressed as :

" some glitters are not gold"

edited
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~(x(glitters(x)gold(x))

u have started from above why don't u use "^" between glitters(x),gold(x))

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“For every person x, if person x is a student in this class then x has studied calculus.”
If S(x) represents the statement that person x is in this class, we see that our statement can be
expressed as ∀x(S(x) → C(x)). [Caution! Our statement cannot be expressed as ∀x(S(x) ∧
C(x)) because this statement says that all people are students in this class and have studied
calculus!]
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i have read this statement in kenneth rosen ..but in this question .. we have to think like this that expect gold some other thing can also glitters that why we dont use "AND" here..??
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x(glitters(x)^gold(x) I think it's meaning is all things are glitters and it is gold.They said to represent all that glitters is gold

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yaar sometimes mathematical logic seems to be difficult ...

??
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+1 vote
The statement “Not all that glitters is gold” can be expressed as follows :

¬(∀x(glitters(x)⇒gold(x)) ... (1)

Where ∀x(glitters(x)⇒gold(x) refers that all glitters is gold. Now ,

∃x¬(glitters(x)⇒gold(x)) ... (2) , Since we know ¬∀x() = ∃x¬()

(Where ∀ refers to -> All and ∃x refers to -> There exists some).

As we know, A⇒B is true only in the case that either A is false or B is true. It can also defined in the other way :

A⇒B=¬A∨B (negationA or B ) ... (3)

From equation (2) and (3) , we have ∃x(¬(¬glitters(x)∨gold(x))

⇒∃x(glitters(x)∧¬gold(x)) ... (4) , Negation cancellation ¬(¬) = () : and ¬(()∨()) = (¬()∧¬()) .

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"Not all that glitters is gold”