Is Every Real Number Has Additive Inverse?’

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.