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

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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

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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

Thank you captainblack. I was wondering from

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(i) Reflexivity: a ~ a

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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

Quote:

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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

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In your notation aRa

RonL