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1)  $\large \Rightarrow \forall x \left (\left ( \bar{p(x)} \wedge q(x) \right )\rightarrow r(x) \right )$

$\large \Rightarrow \forall x \left ( p(x)   \vee \bar{q(x)} \vee r(x) \right )$                 .... 1

$\large \Rightarrow \forall x \left ( p(x)   \vee q(x)  \right )$                                    ......2

and conclusion is 

$\large \Rightarrow \forall x \left ( r(x)   \vee p(x)  \right )$

Now for the above argument to be invalid premises (1 and 2) must be True and conclusion must be False.

There is only one case which leads to conclusion being false

When $\large \forall x $ $r(x)$ is false and $\large \forall x $ $p(x)$ is false. Assuming this let's try to make premises True. Now this is not possible because if you want to make 1st premise True then q(x) must be False which makes 2nd premise False. So both the premise can't be True at the same assumption. Hence this argument is VALID

 

2) $\large \Rightarrow \forall x \left ( \bar{p(x)}   \vee \left (q(x) \wedge r(x) \right ) \right )$                 .... 1

$ \large\Rightarrow \exists x \left ( p(x) \wedge s\left ( x \right ) \right )$                                         .......2

and conclusion is 

$\large \Rightarrow \exists x \left ( r(x)   \wedge s(x)  \right )$

Now conclusion False means that there doesn't exists any x which satisfies both r(x) and q(x). Now to make Premise True.

Second premise is true that means there exist an x which satisfies both p(x) and s(x). From this we extracted that $\exists x$ for which s(x) is true. So to make conclusion false r(x) must be false, and to make premise true we need to make sure that

$\large \forall x \left ( (q(x) \wedge r(x) \right )$  is true because $\large  \forall x \left ( \bar{p(x)}    \right )$  cannot be True. So to make $\large \forall x \left ( (q(x) \wedge r(x) \right )$ True we are saying that for all x r(x) is true which makes conclusion True and it is impossible to make premise True and Conclusion False at the same assumption. Hence it is Valid.

Answer is (C) Both arguments are Valid

 

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