We define the relation of divisibility on by a b iff k b k

We define the relation of divisibility on by a \\ b iff k: b = ka. Show that | is reflexive, anti-symmetric, and transitive.

Solution

Reflexive:

proof:

a = 1*a

=>
a|a

=>
| is reflexive

anti symmetric:
let a|b and b|a

then b = am, a=bn for some integers m,n

=>
b= bmn

=>
mn = 1

=>
m = n =1

=>
a = b

=>
| is anti symmetric

Transitive:

let a|b, b|c

=>
b = am, c = bn for some integers m,n

=>
c = bn = amn = a(mn)

=>
a|c

=>
| is transitive

thus proved