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.