a. Given a labeled directed graph (N, E, l) with N a set of vertices, E ? N x N a set of edges, and l: E ? L a function assigning labels from a set L to edges. Let source and target be functions on E such that source(s, t) = s and target(s, t) = t. Formally formulate the following properties:

i. There are no nodes that are target of more than two edges with identical labels.

ii. There is at least one path of length three where all the edges have identical labels.

b. Does a binary operation * exist such that (N, +) and (Z, *) are isomorphic? Prove your answer.

Note:

N is the set of natural numbers and Z is the set of integers. Two algebraic structures (A, x) and are isomorphic if and only if a bijection h : A ? B exists such that

(B, 0) h(x) x h(y) = h(x & y).