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Equivalence mapping from integers to rationals

  • #1
784
11

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?
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618

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.
 
Last edited:
  • #3
784
11
Oh. right, Thanks dood u da best
 

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