Show that a complete digraph with n nodes has the maximum number of edges if n is not equal to 1. Show that the digraphs given below are isomorphic. Obtain the adjacency matrix A o... Show that a complete digraph with n nodes has the maximum number of edges if n is not equal to 1. Show that the digraphs given below are isomorphic. Obtain the adjacency matrix A of the digraph given in the figure. Find the elementary paths of length 1 and 2 from v1 to v0. Show that there is also a simple path of length 4 from v0 to v0. Prove that every infinite set is equivalent to one of its proper subsets. Define closed sets with example. Define connected sets with examples. Prove that a set E in R is connected if E is one of the intervals. Define direct product of lattices with example. What is subAlgebra with example? Define directed graph, simple graph, weighted graph, and define a directed tree.
Understand the Problem
The question is asking to prove several properties related to directed graphs, adjacency matrices, and boolean matrices, as well as definitions of specific graph types and sets. It involves mathematical proofs and definitions in the context of graph theory and set theory.
Answer
Maximum edges in a complete digraph: $E = n(n  1)$.
Answer for screen readers

Maximum number of edges in a complete digraph with $n$ nodes: $E = n(n  1)$.

To show isomorphism of the given digraphs, verify the onetoone correspondence of edges.

Construct the adjacency matrix $A$, then verify paths using matrix multiplications.

Boolean matrix identity: $ (I  A)A^k = (I  A)A^{k1} $

Every infinite set $S$ is equivalent to one of its proper subsets.
Steps to Solve
 Proving the maximum edges in a complete digraph
A complete digraph with $n$ nodes has edges between every pair of nodes. Since each edge can be directed in two ways, the maximum number of edges is calculated as: $$ E = n(n  1) $$ This equation considers that each of the $n$ nodes can connect to $n  1$ other nodes, without loops.
 Showing isomorphism of the given digraphs
To show the two given digraphs are isomorphic, first identify the vertex sets and their connections in both diagrams.
 List the vertices of the first digraph along with their outgoing edges.
 Then, do the same for the second digraph.
If there is a onetoone correspondence between vertex sets such that the edge connections are preserved, the digraphs are isomorphic.
 Obtaining the adjacency matrix
The adjacency matrix $A$ for a digraph is defined such that $A_{ij} = 1$ if there's a directed edge from vertex $V_i$ to vertex $V_j$, and $0$ otherwise. For each pair of vertices, update the matrix elements based on the edges present in the digraph.
For example, if there's an edge from $V_1$ to $V_2$, then $A_{12} = 1$, otherwise, $A_{12} = 0$.
 Elementary paths verification
To find elementary paths from vertex $V_1$ to $V_2$, verify using the adjacency matrix whether paths exist using matrix multiplication:
 Compute $A^2$, which shows paths of length 2.
 Compute $A^3$, which shows paths of length 3.
 Check whether the entries $(1,2)$ indicate a path of length 1 or greater from $V_1$ to $V_2$.
This confirms whether there's a path of length 1 or 2.
 Verifying for higher powers of the adjacency matrix
Calculate $A^k$ for larger values of $k$ to find paths of longer lengths. This allows verification for paths between vertices. A positive entry at position $(i,j)$ indicates that there is a path from $V_i$ to $V_j$ of at least length $k$.
 Boolean matrix identity proof
To prove the matrix equation, start with the left side and manipulate it stepbystep according to matrix identities: $$ (I  A)A^k = (I  A)A^{k1} $$ Use the identity matrix properties and apply suitable simplifications to reach the right side.
 Subsets of infinite sets
To prove that every infinite set $S$ is equivalent to one of its proper subsets, you can use Cantor's theorem, which states that there is always a way to pair elements of the infinite set with elements of a proper subset, demonstrating that both sets have the same cardinality.

Maximum number of edges in a complete digraph with $n$ nodes: $E = n(n  1)$.

To show isomorphism of the given digraphs, verify the onetoone correspondence of edges.

Construct the adjacency matrix $A$, then verify paths using matrix multiplications.

Boolean matrix identity: $ (I  A)A^k = (I  A)A^{k1} $

Every infinite set $S$ is equivalent to one of its proper subsets.
More Information
 A complete digraph maximizes edge connections, illustrating the concept of full connectivity in directed graphs.
 Isomorphic graphs maintain structural properties, important for equivalency in graph theory.
 The adjacency matrix provides a numerical representation of graph connectivity.
 Sets and their cardinalities are key concepts in understanding infinities in mathematics.
Tips
 Forgetting to check for loops in the complete digraph calculation can lead to incorrect edge counts.
 Confusing isomorphic graphs with equivalent graphs; isomorphic means identical structure, while equivalent implies just similar size.
 Misunderstanding how to calculate powers of the adjacency matrix can lead to errors in path verification.