Question

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

Answer 1

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