Thread: 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?

Originally Posted by oldguy

Question:

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

Is it reflexive,

obviously, for all we have , thus the relation is reflexive

is it symmetric

for , if does it mean that ?

is it antisymmetric

for , if and does it mean that ?

is it transitive?

for . if and , does it mean that ?

Hello, oldguy!

The relation on where means

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

For any natural numnber
. . That is: .
Hence, is reflexive.

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

If and , then
Hence, is antisymmetric.

If and , then . **
Hence, is transitive.


~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

** Proof of transitivity

If , then: . for some integer

If , then: . for some integer

Then: .

Therefore: .