Equivalence mapping from integers to rationals

[PsychonautQQ]

Homework Statement

Let * and = be defined by a*b means a - b is an element of the integers and a = b means that a - b is an element of the rationals. Suppose there is a mapping P: (* equivalence classes over the real numbers) --> (= equivalence classes over the real numbers). show that this mapping is onto and well defined.

Homework Equations

None.

The Attempt at a Solution

I'm confused, wouldn't this mapping NOT be onto? I mean, if you take all the equivalence classes defined by * it couldn't cover all the equivalence classes covered by =, since = deals with rationals and * integers. Is this a misprint in the book or am I mistaken?

 

[Dick]

Homework Statement

Let * and = be defined by a*b means a - b is an element of the integers and a = b means that a - b is an element of the rationals. Suppose there is a mapping P: (* equivalence classes over the real numbers) --> (= equivalence classes over the real numbers). show that this mapping is onto and well defined.

Homework Equations

None.

The Attempt at a Solution

I'm confused, wouldn't this mapping NOT be onto? I mean, if you take all the equivalence classes defined by * it couldn't cover all the equivalence classes covered by =, since = deals with rationals and * integers. Is this a misprint in the book or am I mistaken?

Mistaken. If Z is the integers and Q is the rationals, then an equivalence class of * is a set of the form r+Z where r is a real number. An equivalence class of = is a set of the form s+Q where s is real. Can't you think of a sort of obvious way to map one onto the other? Then try and prove your map is well defined and onto. You can't really prove a map is anything until you define it.

 

[PsychonautQQ]

Oh. right, Thanks dood u da best