1. ## 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?

2. Originally Posted by oldguy
Question:

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

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

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

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

is it transitive?
for $\displaystyle a,b,c \in \mathbb{N}$. if $\displaystyle a|b$ and $\displaystyle b|c$, does it mean that $\displaystyle a|c$?

3. Hello, oldguy!

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

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

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

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

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

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

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** Proof of transitivity

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

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

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

Therefore: .$\displaystyle a|c.$