Problem in understanding the meaning

[thudda]

What is the difference between the 2 expressions

1) for all x belongs to ℝ there exists y belongs to ℝ such that f(x)=y
2) there exists y belongs to ℝ such that for all x belongs to ℝ , f(x)=y

I want to know the exact difference.

 

[tiny-tim]

hi thudda! welcome to pf!

tell us what you think, and then we'll comment!

 

[thudda]

Hi..thanks..:)

I think the 1st expression implies that for every x value there is a corresponding y value.And the 2nd imply that for all x values there's one or a set of y values...what I want to know is whether it is one y value or a set of y values..

 

[tiny-tim]

hi thudda!

I think the 1st expression implies that for every x value there is a corresponding y value.

yup … basically, it doesn't say anything more than that f is a function!

2) there exists y belongs to ℝ such that for all x belongs to ℝ , f(x)=y

And the 2nd imply that for all x values there's one or a set of y values...what I want to know is whether it is one y value or a set of y values..

ah, it's one y value …

"there exists y" always means there exists a y

(btw, when i see $y\in R$, i always read that as "y in R" … it's shorter, and i think, easier, than "y belongs to R" )

 

[pasmith]

"there exists y" always means there exists a y

Without more, both "there exists a(n)" and its abbreviation $\exists$ mean "there exists at least one". If you want to specify uniqueness, you must do so expressly. The abbreviation for "there exists exactly one" is $\exists !$ - the usual symbol followed by an exclamation mark.

As to the OP's examples: The first isn't quite the definition of a function; it would be if it asserted that y was unique. In the second, it follows from the definition of a function that if such a y exists then it must be unique.