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Plese tell me the proper approach to solve these kind of questions.

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Answer : B

$S_1 : $ $\forall x \exists y \forall z(x + y = z)$, Where domain is Real Numbers.

It is Saying that For each real number, say $x = 5$, there is some real number $y$ such that $x+ y = z$, for all real numbers $z$. 

This is False as it says that $5 + y = ALL \,\,Real\,\,Numbers$, Where $y$ is some fix real number. This is obviously wrong. 

$S_2 : $  $\exists x \forall y \exists z(x + y = z)$, Where domain is Real Numbers.

 It says that There is some Real number, say $x$, such that for each real numbers $y$, $x + y = z$ for some real number $z$(may be different values of $z$ for different values of $y$). 

It is True. As Say you take $x = 5$, then You can add Any Real number $y$ to it and It will result in some real number(not necessarily same for each $y$) 


Note : Actually,  $S_2$ is a subset condition of the following formula :

$\forall x \forall y \exists z(x + y = z)$, Where domain is Real Numbers. Which is True as For Every Pair of real numbers, say $(5,7)$, we do have Some Real number $z$ such that $5+7 = z$, here, for pair $(5,7)$, $z$ is 12. For different pairs $(x,y)$, $z$ may be different but it will surely exist.

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