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):6|n-m\\}"

It is equivalence relation.

Reflexivity: since "m-m=0" and "6|0" , then "m(mod6)=m"

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

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

Since "k-m=(k-n)+(n-m)" we have "6|(k-m)" and "m=k(mod6)"

Partition is:

"[m]=\\{n-6a\\}" , where "a\\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<b" then "b>a"

Transitivity: if "a<b" and "b<c" then "a<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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