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: 54

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: 17,124
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'
'You made a human being happy! There is no further achievement.' -bobbym

Offline

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

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

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