Which of the following relations are equivalence relations E

Which of the following relations are equivalence relations? Either prove all three conditions hold or give a counter example for one of the conditions.

(a) Let S be the set of all sequences of 0s 1s and 2s of length 3. Define a relation R by xRy if and only if the sum of the entries in the sequences in x and y are equal. So (1, 2, 1) is related to (2, 2, 0) since the sum of the entries is 4.

For each of the relations that are equivalence relations give (either list or describe) the equivalance classes.

Solution

(a)It is an equivalent relation

sum of elements in x = sum of element in x => xRx => Reflexive

if xRy => sum of elements in x = sum of elements in y

=>
sum of elements in y = sum of elements in x => yRx => symmetric

let xRy, yRz =>

sum of elements in x = sum of elements in y,
sum of elements in y = sum of elements in z

sum of elements in x = sum of elements in z => xRz => transitive

=>

the above three properties prove it is an equivalence relation

thus proved

(b)

NOT an equivalence relation

counter example :

1-1 =0 not an odd number => 1 is not related to 1 => NOT reflexive => NOT equivalent

(c)

NOT an equivalent relation

counter example :

|1-5| <=5

=> 1R5

|5-9| <=5

=>

5R9

|1-9| = 8

=>

1 is not related to 9 => NOT transitive => NOT equivalent

(d)

x^2 + y^2 = x^2 +y^2

=>

(x,y) R (x,y) => it is reflexive

let (x,y)R(a,b)

=>

x^2 + y^2 = a^2 + b^2

=>

a^2 +b^2 = x^ 2+y^2

=>
(a,b) R (x,y) => it is symmetric

let (x,y) R(a,b) and (a,b) R(p,q)

=>

x^2 +y^2 = a^2 +b^2

a^2 +b^2 = p^2 + q^2

=>

x^2 +y^2 = p^2 +q^2

=>
(x,y)R(p,q) => it is transitive

the above three properties prove that it is equivalent relation