edited by
6,524 views
37 votes
37 votes

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. $\forall x: \text{glitters} (x)\Rightarrow \neg \text{gold}(x)$
  2. $\forall x:\text{gold} (x)\Rightarrow \text{glitters}(x)$
  3. $\exists x: \text{gold}(x)\wedge \neg \text{glitters}(x)$
  4. $\exists x: \text{glitters}(x)\wedge \neg \text{gold}(x)$
edited by

6 Answers

Best answer
35 votes
35 votes

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

$\neg(\forall x(\text{glitters}(x)\implies \text{gold}(x)))$ 

(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 

 $\exists x(\text{glitters}(x)\wedge  \neg \text{gold}(x))$

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

So option (D) is correct. 

edited by
16 votes
16 votes

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
0 votes
0 votes
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 ¬(()∨()) = (¬()∧¬()) .

So Answer is (D) .
Answer:

Related questions

37 votes
37 votes
11 answers
2