Which of the following relations are equivalence relations? Either prove all thr
ID: 3416998 • Letter: W
Question
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.
Explanation / Answer
(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
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