(2) The following statement is true: If G = (V, E) is a graph, |V| = n, and |E| < n - 1, then G

Step 1: Identify the statement to be analyzed. The statement in question is: "If G = (V,E) is a

Question

(2) The following statement is true: If G = (V, E) is a graph, |V| = n, and |E| < n - 1, then G is not connected. The following "proof" is wrong. What is wrong with it? Remember, your goal is not to write a correct proof. You goal is to simply identify what is wrong with this one. Proposed proof: A graph G = (V, E) is connected if for every pair of vertices u, v ? V, u is connected to v. Suppose G = (V, E) is a graph, |V| = n, and |E| < n - 1. Consider any vertex v ? V(G). Since there are fewer than n - 1 edges in G, the degree of v is less than n - 1. So v has fewer than n - 1 neighbors. Noting that |V - {v}| = n - 1, we find that there is at least one vertex of V - {v} which is not connected to v. So G is not connected.

          (2) The following statement is true: If G = (V, E) is a graph, |V| = n, and |E| < n - 1, then G is not connected.
The following "proof" is wrong. What is wrong with it? Remember, your goal is
not to write a correct proof. You goal is to simply identify what is wrong with this
one.
Proposed proof: A graph G = (V, E) is connected if for every pair of vertices
u, v ? V, u is connected to v. Suppose G = (V, E) is a graph, |V| = n, and
|E| < n - 1. Consider any vertex v ? V(G). Since there are fewer than n - 1
edges in G, the degree of v is less than n - 1. So v has fewer than n - 1 neighbors.
Noting that |V - {v}| = n - 1, we find that there is at least one vertex of V - {v}
which is not connected to v. So G is not connected.

(2) The following statement is true: If G = (V, E) is a graph, |V| = n, and |E| < n - 1, then G is not connected.
The following "proof" is wrong. What is wrong with it? Remember, your goal is
not to write a correct proof. You goal is to simply identify what is wrong with this
one.
Proposed proof: A graph G = (V, E) is connected if for every pair of vertices
u, v ? V, u is connected to v. Suppose G = (V, E) is a graph, |V| = n, and
|E| < n - 1. Consider any vertex v ? V(G). Since there are fewer than n - 1
edges in G, the degree of v is less than n - 1. So v has fewer than n - 1 neighbors.
Noting that |V - v| = n - 1, we find that there is at least one vertex of V - v
which is not connected to v. So G is not connected.

Added by Adrian W.

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