Let n be a positive integer and let Sn be any set with |Sn|= n. Define Dn to be the digraph with V (Dn)= P (Sn) , the set of all subsets of Sn where (X, Y) element of A(Dn) if and only if X contains Y properly as a subset. a) Make a pictorial representation of D3 b) Prove that D has a unique source. c) Prove that Dn has a unique sink.
Let n be a positive integer and let Sn be any set with |Sn|= n. Define Dn to be the digraph with V (Dn)= P (Sn) , the set of all subsets of Sn where (X, Y) element of A(Dn) if and only if X contains Y properly as a subset. a) Make a pictorial representation of D3 b) Prove that D has a unique source. c) Prove that Dn has a unique sink.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let n be a positive integer and let Sn be any set with |Sn|= n. Define Dn to be the digraph with V (Dn)= P (Sn) , the set of all subsets of Sn where (X, Y) element of A(Dn) if and only if X contains Y properly as a subset.
a) Make a pictorial representation of D3
b) Prove that D has a unique source.
c) Prove that Dn has a unique sink.
d) Find a necessary and sufficient condition for Dn to have carrier vertices.
e) Find a formula for the size Dn in terms of n.
f) Prove that D has no circuit.
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