# Relations

• Dec 1st 2007, 06:48 PM
oldguy
Relations
Question:

The relation R on {1,2,3,....} where aRb means a|b.

Is it reflexive, is it symmetric, is it antisymmetric, is it transitive?
• Dec 1st 2007, 07:18 PM
Jhevon
Quote:

Originally Posted by oldguy
Question:

The relation R on {1,2,3,....} where aRb means a|b.

Is it reflexive,

obviously, for all $a \in \mathbb{N}$ we have $a|a$, thus the relation is reflexive

Quote:

is it symmetric
for $a,b \in \mathbb{N}$, if $a|b$ does it mean that $b|a$?

Quote:

is it antisymmetric
for $a,b \in \mathbb{N}$, if $a|b$ and $b|a$ does it mean that $a = b$?

Quote:

is it transitive?
for $a,b,c \in \mathbb{N}$. if $a|b$ and $b|c$, does it mean that $a|c$?
• Dec 1st 2007, 07:25 PM
Soroban
Hello, oldguy!

Quote:

The relation $R$ on $\{1,2,3, \cdots\}$ where $a\text{R}b$ means $a|b.$

Is it reflexive? .Symmetric? .Antisymmetric? .Transitive?

For any natural numnber $a,\;a \div a \:=\:1$
. . That is: . $a|a$
Hence, $\text{R}$ is reflexive.

If $a$ divides $b$, it does not follow that $b$ divides $a.$
Hence, it is not symmetric.

If $a|b$ and $b|a$, then $a = b.$
Hence, $\text{R}$ is antisymmetric.

If $a|b$ and $b|c$, then $a|c.$ . **
Hence, $\text{r}$ is transitive.

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** Proof of transitivity

If $a|b$, then: . $b \:=\:ma$ for some integer $m.$

If $b|c$, then: . $c \:=\:nb$ for some integer $n.$

Then: . $c \:=\:nb \:=\:n(ma) \:=\:(mn)a$

Therefore: . $a|c.$