Let D be the set of all set of all finite subsets of positive integers, and define T : Z + → D by the following rule: For every integer n , T ( n ) = the set of all of the positive divisors of n.

1. Is T one-to-one? Prove or give a counterexample.

Is T onto? Prove a give counterexample.

(a)

To check:

Whether T is one-to-one or not.

Given information:

Consider D be the set of all finite subsets of positive integers, and T:ℤ+→D is defined by the rule: For all integers n, T(n)= the set of all of the positive divisors of n.

Concept used:

In one-to-one function, distinct elements in domain are mapped with distinct elements in co-domain.

Calculation:

Claim: T is a one-to-one function.

For this consider x,y∈ℤ+ and x≠y. Without loss of generality assume that x<y

(b)

To check:

Whether T is onto or not.