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Given below are two premises with four conclusions drawn from them. Which of the following conclusions could be validly drawn from the premises?

Premises:

  1. No paper is pen
  2. Some paper are handmade

Conclusions:

  1. All paper are handmade
  2. Some handmade are pen
  3. Some handmade are not pen
  4. All handmade are paper
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Let $x$ : An object

      $p(x)$: $x$ is a paper 

      $n(x)$: $x$ is a pen  and

      $h(x)$: $x$ is handmade

 

No paper is pen.$\equiv$ There does not exist an $x$ such that  $x$ is a paper and it is a pen $\equiv\ \sim \exists x ( p(x) \wedge n(x)) \equiv \forall x$$(\sim p(x)\  \vee  \sim n(x) )$

Also from $\forall x (\sim p(x)\  \vee  \sim n(x) )$ we can conclude $\exists x (\sim p(x)\  \vee  \sim n(x) ) \equiv \exists x ( p(x) \rightarrow \sim n(x)) $

 

Some paper are handmade$\equiv$ There exists some $x$ such that $x$ is a paper and it is handmade $\equiv \exists x(p(x) \wedge  h(x))$

 

Using the above two equations

$\exists x ( p(x) \rightarrow \sim n(x)) $

$\underline{\exists x(p(x) \wedge  h(x))}$

$\exists x( \sim n(x) \wedge h(x))$

 

  1. All paper are handmade. $\equiv \forall x$( if $x$ is a paper then it is handmade) $\equiv \forall x ( p(x) \rightarrow h(x)) \equiv \forall x (\sim p(x)\ \vee h(x))$
  2. Some handmade are pen$\equiv$ There exists some $x$ such that $x$ is a handmade and it is a pen $\equiv \exists x(h(x) \wedge  n(x))$
  3. Some handmade are not pen$\equiv$ There exists some $x$ such that $x$ is a handmade and it is not a pen $\equiv \exists x(h(x) \wedge \sim n(x))$
  4. All handmade are paper $\equiv \forall x$( if $x$ is a handmade then it is a paper) $\equiv \forall x ( r(x) \rightarrow f(x)) \equiv \forall x (\sim h(x)\ \vee p(x))$

 

$\therefore$ Option $C.$ is correct

Answer:

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