1. Let W, U, and S be graphs defined as follows: • V(W) is the set of countries in the world; • V(U) is the set of countries in the European Union; V(S) is the set of countries in the Schengen Area; ● for X = {W,U,S}, E(X) is the set of pairs of countries in V(X) that share a land border. Recall that land borders between countries in the Schengen Area are special in that they can be crossed without a passport. (a) The notions of a country and a land border are somewhat ambiguous. Explain the notions you will use to get a precise definition of the graphs W, U, and S. (b) Is S a subgraph of U? Is U an induced subgraph of W? Justify your answers. (c) Using non-mathematical language, explain what it means for a country x if VEV(S) and dw (v) = 0. Give all such countries. Let A = {v Є V(W) \V(S) such that |Nw(v)| > 0 and Nw (v) ≤ V(S)}. (d) Using non-mathematical language, explain what the set A represents in terms of countries and land borders. Give a specific element of A or explain why A is empty. You are planning a holiday that involves a single return flight to an airport and a road trip using a rental car that you pick up and return at that airport. There are two further complications: the car rental company will keep your passport as a deposit, so you cannot use it during the road trip; and your travel companion is easily bored, so you cannot visit a country more than once except when this is required to return the rental car. (e) Give a property of the graphs W, U, and S that is equal to the largest number of countries that can be visited during such a holiday. You do not actually have to determine this number.
1. Let W, U, and S be graphs defined as follows: • V(W) is the set of countries in the world; • V(U) is the set of countries in the European Union; V(S) is the set of countries in the Schengen Area; ● for X = {W,U,S}, E(X) is the set of pairs of countries in V(X) that share a land border. Recall that land borders between countries in the Schengen Area are special in that they can be crossed without a passport. (a) The notions of a country and a land border are somewhat ambiguous. Explain the notions you will use to get a precise definition of the graphs W, U, and S. (b) Is S a subgraph of U? Is U an induced subgraph of W? Justify your answers. (c) Using non-mathematical language, explain what it means for a country x if VEV(S) and dw (v) = 0. Give all such countries. Let A = {v Є V(W) \V(S) such that |Nw(v)| > 0 and Nw (v) ≤ V(S)}. (d) Using non-mathematical language, explain what the set A represents in terms of countries and land borders. Give a specific element of A or explain why A is empty. You are planning a holiday that involves a single return flight to an airport and a road trip using a rental car that you pick up and return at that airport. There are two further complications: the car rental company will keep your passport as a deposit, so you cannot use it during the road trip; and your travel companion is easily bored, so you cannot visit a country more than once except when this is required to return the rental car. (e) Give a property of the graphs W, U, and S that is equal to the largest number of countries that can be visited during such a holiday. You do not actually have to determine this number.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 34EQ
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Transcribed Image Text:1. Let W, U, and S be graphs defined as follows:
• V(W) is the set of countries in the world;
• V(U) is the set of countries in the European Union;
V(S) is the set of countries in the Schengen Area;
● for X = {W,U,S}, E(X) is the set of pairs of countries in V(X) that share a
land border.
Recall that land borders between countries in the Schengen Area are special in that
they can be crossed without a passport.
(a) The notions of a country and a land border are somewhat ambiguous. Explain
the notions you will use to get a precise definition of the graphs W, U, and S.
(b) Is S a subgraph of U? Is U an induced subgraph of W? Justify your answers.
(c) Using non-mathematical language, explain what it means for a country x if
VEV(S) and dw (v) = 0. Give all such countries.
Let A = {v Є V(W) \V(S) such that |Nw(v)| > 0 and Nw (v) ≤ V(S)}.
(d) Using non-mathematical language, explain what the set A represents in terms
of countries and land borders. Give a specific element of A or explain why A
is empty.
You are planning a holiday that involves a single return flight to an airport and
a road trip using a rental car that you pick up and return at that airport. There
are two further complications: the car rental company will keep your passport as
a deposit, so you cannot use it during the road trip; and your travel companion
is easily bored, so you cannot visit a country more than once except when this is
required to return the rental car.
(e) Give a property of the graphs W, U, and S that is equal to the largest number
of countries that can be visited during such a holiday. You do not actually
have to determine this number.
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