1. ## 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

2. 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

3. Thank you captainblack. I was wondering from

(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

4. 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