5.
(a) Describe Warshall's algorithm for the sequences of the matrices $W_k = \begin{bmatrix} w_{ij}^{(k)} \end{bmatrix}$ with
$k \in \{1, 2, ..., n\}$ in terms of pseudocode.
(b) Show that Warshall's algorithm requires $2n^3$ bits operations to compute $W_n = M_R$ from the input $W = M_R$
(c) Let R be the relation defined on a set A = {a, b, c, d, e} by
$R = \{(a, c), (b, d), (c, a), (d, b), (e, d)\}$.
Use Warshall's algorithm to find the transitive closures of R.
Suppose that the boolean matrix $W_k = \begin{bmatrix} w_{ij}^{(k)} \end{bmatrix}$ has 1 in