What is the set \frac{\mathbb{Z}}{2\mathbb{Z}}?

Krzychau1

Krzychau1

Answered question

2022-01-13

What is the set Z2Z?

Answer & Explanation

Dawn Neal

Dawn Neal

Beginner2022-01-14Added 35 answers

Step 1
Fix nN
The set
ZnZ={xZx¬{}nZ}
where
nZ={nyZyZ}
The group
ZnZ={x+nZxZ}
where nZ is as above, is uderstood as a group under the operation
(a+nZ)+(b+nZ)+=(a+b)+nZ
It is a quotient group. Each of its elements is what is known as a coset of nZ
Your case, of course, is when n=2
scomparve5j

scomparve5j

Beginner2022-01-15Added 38 answers

Step 1
Z be the set of all integers.
We want to put all integers into n distinct buckets and two integers belong to same bucket if they are similar.
Now, it is meaningless until we define what does it mean by the word ''similar '' and ''bucket''.
Two integers a and b are similar (ab) if ab is divisible by n.
It is easy to check that the relation is an equivalence relation.
Then the equivalence class(bucket) of a,
[a]={bZ:ba}
aZ and n>1 by division algorithm,
a=nq+r (q: quotient, r: reminder, 0r(n1))
ar=nq and hence, ar
[a]=[r]
Hence, there is n distinct equivalence classes
[0],[1],,[n1]
Then the set of all equivalence classes
{[0],[1],,[n1]}=ZnZ
In your question, n=2
Z2Z={01}
Two integers belongs to same equivalence class iff they leave the same reminder upon division by 2.
Like, 2[0] because both are divisible by 2.

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