Distinguish between two statements
$$(\forall x \in R)\ ( \exists y \in R)\ (x + y = 0 )\ and\ ( \exists y \in R)\ ( \forall x \in R)\ (x + Y = 0)$$

Solution:

Here

$(\forall X \in R)( \exists Y \in R) (X + Y = 0 )$

This statement is to mean that for every real number has a additive inverse this statement is true.

$(\exists y \in R)(\forall x \in R) (x + Y = 0)$

Is not true, since no y will be additive inverse for every real numbers.

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