# Equivalence relation

• May 19th 2007, 05:57 PM
smoothi963
Equivalence relation
Determine whether the following relation is an equivalence relation:

1. For all x,y element of Real # xRy <-> x-y is an integer.

2. For all a,b elemnt of Natural # aRb <-> a ends in the same digit in which be ends

With equivalence relation, i am so lost. I could really use some help here please. This section of the book is just not making much sense to me. Thank you
• May 20th 2007, 12:16 AM
CaptainBlack
Quote:

Originally Posted by smoothi963
Determine whether the following relation is an equivalence relation:

1. For all x,y element of Real # xRy <-> x-y is an integer.

2. For all a,b elemnt of Natural # aRb <-> a ends in the same digit in which be ends

With equivalence relation, i am so lost. I could really use some help here please. This section of the book is just not making much sense to me. Thank you

For a relation to be an equivalence relation on X, for any a, b and c in X the
following must hold:

(i) Reflexivity: a ~ a
(ii) Symmetry: if a ~ b then b ~ a
(iii) Transitivity: if a ~ b and b ~ c then a ~ c.

1) X = R, let a, b and c be in R.

Clearly aRa <-> a-a is an integer but a-a=0 which is an integer so (i) holds

Also if a-b is an integer then so is b-a, so (ii) holds

Now if a-b is an integer, and so is b-c then:

a-c = (a-b) + (b-c)

is the sum of two integers and so is an iteger so aRc, so (iii) holds

Hence we conclude that R is an equivalence relation on R.

2) Same as above just go through checking reflexivity, symmetry and
transitivity if the hold the R is an equivelence relation if not it is not

RonL
• May 20th 2007, 04:07 AM
smoothi963
Thank you captainblack. I was wondering from

Quote:

(i) Reflexivity: a ~ a
what does a ~ a mean? I don't know what the ~ represent. I am jsut trying to understand this so i can do well on my test. Thank you
• May 20th 2007, 04:53 AM
CaptainBlack
Quote:

Originally Posted by smoothi963
Thank you captainblack. I was wondering from

what does a ~ a mean? I don't know what the ~ represent. I am jsut trying to understand this so i can do well on my test. Thank you