Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2013-04-10 15:06:37

mrpace
Member
Registered: 2012-08-16
Posts: 27

Show that ~ is an equivalence relation

so yea i'm not really understanding what an equivalence relation is even. Can anyone do this problem?

Let ~ be the relation defined on Z by
m~n <--> 2 devides m+n

show that ~ is an equivalence relation
describe the partition of Z determined by the equivalence classes of ~

any help is much appreciated.
thanks.

Offline

#2 2013-04-10 15:34:03

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 14,235
Website

Re: Show that ~ is an equivalence relation

1. m~m since 2|m+m = 2m (Thats reflexivity)
2. Let us take for granted m~n which means 2|m+n.
    Again, n~m, means 2|n + m. Now m+n=n+m
    Therefore, IF m~n, THEN n~m (Thats symmetry)
3. Let us take for granted m~n and n~o
    Therefore, 2|m+n and 2|n+o. Let m + n = 2k and n + o = 2j, Now m + o = 2k + 2j - 2n = 2(k+j -n)
    Thus 2|m+o; Thus, m~o (Thats Transitivity)

From 1, 2 and 3 ~ is symmetrical, transitive and reflexive. Therefore ~ is an equivalence relation


smile


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

Offline

#3 2013-04-10 20:58:40

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,819

Re: Show that ~ is an equivalence relation

And, you will notice that x~y if and only if x and y are of the same parity, so the classes of equivalence in Z are the even integers and the odd integers.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

Board footer

Powered by FluxBB