Question
Let be a fixed positive integer. Define a relation on as follows:
divides .
Show that is an equivalence relation on .
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Let $$n$$ be a fixed positive integer.
Symmetry: Let $$a,b\in Z$$ be such that
$$(a,b)\in R$$
$$\Rightarrow$$ $$n$$ divides $$a-b$$
$$\Rightarrow$$ $$a-b=n\lambda$$ for some $$\lambda \in Z$$
$$\Rightarrow$$ $$b-a=n(-\lambda)$$, where $$-\lambda \in Z$$
$$\Rightarrow$$ $$n$$ divides $$b-a\Rightarrow (b,a)\in R$$
Thus $$(a,b)\in R\Rightarrow (b,a)\in R$$. So, $$R$$ is a symmetric relation on $$Z$$.
Transitivity. Let $$a,b,c\in Z$$ be such that $$(a,b)\in R$$ and $$(b,c)\in R$$. Then,
$$(a,b)\in R\Rightarrow n$$ divides $$a-b\Rightarrow a-b=n\lambda$$ for some $$\lambda \in Z$$
and $$(b,c)\in R\Rightarrow n$$ divides $$b-c\Rightarrow b-c=n\mu$$ for some $$\mu \in Z$$
$$\therefore$$ $$a-b+b-c=n(\lambda +\mu)$$
$$\Rightarrow$$ $$a-c=n(\lambda +\mu)$$, where $$\lambda +\mu \in Z$$
$$\Rightarrow$$ $$n$$ divides $$a-c$$
$$\Rightarrow$$ $$(a,c)\in R$$
thus, $$(a,b)\in R$$ and $$(b,c)\in R\Rightarrow (a,c)\in R$$
So, $$R$$ is a transitive relation on $$Z$$
Hence, $$R$$ is an equivalence relation on $$Z$$.
Let the relation be $$R$$ on $$Z$$ is given by $$R=\left\{ \left( a,b \right) :n\quad \text{divides}\quad a-b \right\} $$.
We observe the following properties of relation $$R$$
Reflexivity: for any $$a\in Z$$
$$a-a=0=0\times n$$
$$\Rightarrow$$ $$n$$ divides $$a-a\Rightarrow (a,a)\in R$$
So, $$R$$ is relexive relation on $$Z$$
We observe the following properties of relation $$R$$
Reflexivity: for any $$a\in Z$$
$$a-a=0=0\times n$$
$$\Rightarrow$$ $$n$$ divides $$a-a\Rightarrow (a,a)\in R$$
So, $$R$$ is relexive relation on $$Z$$
Symmetry: Let $$a,b\in Z$$ be such that
$$(a,b)\in R$$
$$\Rightarrow$$ $$n$$ divides $$a-b$$
$$\Rightarrow$$ $$a-b=n\lambda$$ for some $$\lambda \in Z$$
$$\Rightarrow$$ $$b-a=n(-\lambda)$$, where $$-\lambda \in Z$$
$$\Rightarrow$$ $$n$$ divides $$b-a\Rightarrow (b,a)\in R$$
Thus $$(a,b)\in R\Rightarrow (b,a)\in R$$. So, $$R$$ is a symmetric relation on $$Z$$.
Transitivity. Let $$a,b,c\in Z$$ be such that $$(a,b)\in R$$ and $$(b,c)\in R$$. Then,
$$(a,b)\in R\Rightarrow n$$ divides $$a-b\Rightarrow a-b=n\lambda$$ for some $$\lambda \in Z$$
and $$(b,c)\in R\Rightarrow n$$ divides $$b-c\Rightarrow b-c=n\mu$$ for some $$\mu \in Z$$
$$\therefore$$ $$a-b+b-c=n(\lambda +\mu)$$
$$\Rightarrow$$ $$a-c=n(\lambda +\mu)$$, where $$\lambda +\mu \in Z$$
$$\Rightarrow$$ $$n$$ divides $$a-c$$
$$\Rightarrow$$ $$(a,c)\in R$$
thus, $$(a,b)\in R$$ and $$(b,c)\in R\Rightarrow (a,c)\in R$$
So, $$R$$ is a transitive relation on $$Z$$
Hence, $$R$$ is an equivalence relation on $$Z$$.
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Question Text | Let be a fixed positive integer. Define a relation on as follows: divides . Show that is an equivalence relation on . |
Answer Type | Text solution:1 |
Upvotes | 150 |