Is the Canonical Map Z to Zsubscript5 1-1 and Onto?

  • Thread starter Thread starter Punkyc7
  • Start date Start date
  • Tags Tags
    Map Proof
Click For Summary
SUMMARY

The canonical map from Z (the set of all integers) to Z5 (the set of equivalence classes modulo 5) is established as onto but not one-to-one. This conclusion arises from the fact that Z encompasses all integers, while Z5 consists of the equivalence classes [0], [1], [2], [3], and [4]. The mapping fails to be one-to-one because multiple integers, such as 0 and 5, map to the same equivalence class [0]. A clear proof demonstrates that for any equivalence class [a] in Z5, there exists an integer in Z that maps to it.

PREREQUISITES
  • Understanding of equivalence classes in modular arithmetic
  • Familiarity with the concepts of one-to-one and onto functions
  • Basic knowledge of integer sets and mappings
  • Proficiency in mathematical proof techniques
NEXT STEPS
  • Study the properties of modular arithmetic and equivalence relations
  • Learn about functions and their classifications, specifically one-to-one and onto functions
  • Explore examples of canonical maps in different mathematical contexts
  • Review proof techniques in set theory and functions
USEFUL FOR

Mathematics students, educators, and anyone interested in understanding mappings between sets, particularly in the context of modular arithmetic and equivalence classes.

Punkyc7
Messages
415
Reaction score
0
Determine if the canonical map Z to Zsubscript5 is 1-1 and onto. Prove your answer


Im not sure how to prove it but I am almost positive that its onto and not 1-1. I believe it onto because Z contains all the integers and Zsubscript5 contain the equivalence classes [0] [1] [2] [3] [4]. I don't believe that its 1-1 because 0 and 5 get mapped onto [0]
 
Physics news on Phys.org
Your proof looks good to me! You might want to be a bit more explicit about being onto. That is, if I give you the equivalence class [a], what integer gets mapped to [a] by the canonical map?
 

Similar threads

Find 10 Terms of the Laurent Expansion of 1/z(1-z)^2 in |z|>1
  • · Replies 7 ·
Replies
7
Views
2K
Prove that |f| is bounded by a quotient
  • · Replies 4 ·
Replies
4
Views
2K
Given surjective ##T:V\to W##, find isomorphism ##T|_U## of U onto W.
  • · Replies 1 ·
Replies
1
Views
2K
Criterion for a positive integer a>1 to be a square
  • · Replies 4 ·
Replies
4
Views
2K
Elements of Aut(Z/4Z) and their Inclusion in Inn(Z/4Z)
  • · Replies 9 ·
Replies
9
Views
3K
Medium Hard continuity proof Tutorial Q6
  • · Replies 2 ·
Replies
2
Views
1K
Prove bijection from (0,1] to (0,1)
  • · Replies 4 ·
Replies
4
Views
2K
Prove that range ##T'## = ##(\text{null}\ T)^0##
  • · Replies 1 ·
Replies
1
Views
1K
Conformal Mapping: Find Points of Z Plane for f(z)=-(1-z)/(1+z)
  • · Replies 7 ·
Replies
7
Views
3K
Mobius transformation for the first quadrant
  • · Replies 2 ·
Replies
2
Views
3K