We define the relation of divisibility on N by a b iff 3k 6

We define the relation of divisibility on N+ by a | b iff 3k 6 N*:b = ka. Show that I is reflexive, anti-symmetric, and transitive.

Solution

Transitive : We have a | b i.e we know b = ka . Let us suppose b | c i.e. c =mb where m belongs to N .

Now c = mb = m * (ka) = (m*k) a = na (Since m and k are integers , we have n as some other integer equal to m*k)

Since c = na , where n belongs to N , we can say a | c. Hence the transitive property is proved , ie. if a | b and b | c ,then a | c.

Anti Symmetric :

A relation R is anti symmetric , if xRy and yRx then x = y . Now let us test for division relation. Now we have a | b i.e . b= ka. Let us suppose b | a i.e. a = mb , then we have , a = m * (ka ) = (m*k) a

So a = (m*k) a, essentially m*k should be 1 , hence m and k are reciprocals to each other .But since both m and k are integers ,both of them should be equal to 1 for their product to be 1. Hence m = k =1.Hence from b= ka ,we get b= 1* a = a

Hence the anti symmetric property is proved i.e if a | b and b | a , then a = b.

Reflexive Property: The reflexive property holds good because every number divides itself .Hence division is reflexive.