a) Let A = (1,2,3,4) and R = ((1 ,2),(2,4),( 1,3 ),(3 ,2)}. Find the transitive closure of R by Warshall’s algorithm.
b) Let A = {a,b,c}. show that (P(A),c ) is a poset and draw its Hasse diagram.
c) Define Chains and Antichains.
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Expert's answer
2020-11-03T17:52:16-0500
(a) The matrix that represents R is shown below. It will be an initial matrix for the algorithm.
W0=⎣⎡0000101010000100⎦⎤
For every step, Wi,jn:=Wi,jn−1∨(Wi,kn−1∧Wk,jn−1) for some k.
In other words, we can add a new pair (i,j) to the relation Ri corresponding to Wi, if and only if there is some k in Ri−1 corresponding to Wi−1.
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