Compute the indicated quantities for the given homomorphism¢. 

Ker(∅) and ∅(4, 6) for∅ : Z x Z → Z x Z where ∅(1, 0) = (2, -3) and ∅(0, 1) = (-1, 5)

Data  from Exercise 46

Let a group G be generated by { a_i | i ∈ I}, where I is some indexing set and a_i ∈ G for all i ∈ I. Let ∅ : G → G' and µ : G → G' be two homomorphisms from G into a group G', such that ∅(a_i) = µ(a_i) for every i ∈ I. Prove that ∅ = µ. [Thus, for example, a homomorphism of a cyclic group is completely determined by its value on a generator of the group.]