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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