Answer to Question #117155 in Discrete Mathematics for Priya

Question #117155
Determine the following relation is an equivalence relation or not. If the relation is an equivalence relation, describe the partition given by it. m~n in Z if m=n mod 6. 51. Which of them are equivalence relations?
(a) "less than" on the set N
(b) "has the same shape as" on the set of all triangles
1
Expert's answer
2020-06-11T18:56:52-0400

1.

We have relation: {(m,n):6nm}\{(m,n):6|n-m\}

It is equivalence relation.

Reflexivity: since mm=0m-m=0 and 606|0 , then m(mod6)=mm(mod6)=m

Symmetry: since nm=(mn)n-m=-(m-n) , then 6(mn)6|(m-n) and n=m(mod6)n=m(mod6)

Transitivity: if m=n(mod6)m=n(mod6) and n=k(mod6)n=k(mod6) then 6(nm),6(kn)6|(n-m),6|(k-n) .

Since km=(kn)+(nm)k-m=(k-n)+(n-m) we have 6(km)6|(k-m) and m=k(mod6)m=k(mod6)

Partition is:

[m]={n6a}[m]=\{n-6a\} , where aNa\in N


2.

a) Relation "less than" on the set N is not equivalence relation:

No reflexivity: a number cannot be less than itself.

No symmetry: if a<ba<b then b>ab>a

Transitivity: if a<ba<b and b<cb<c then a<ca<c


b) Relation "has the same shape as" on the set of all triangles is equivalence relation:

Reflexivity: one triangle has single shape

Symmetry: two triangles can have same shape

Transitivity: three triangles can have same shape


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