Symmetric relation v.s. symmetric matrix

Here's the LONG question...

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A *graph* is a finite set of *vertices* *V* along with a relation *E* on *V*. The elements of the relation *E* are called *edges* (and the vertices are sometimes called *nodes*). We think of the edges as linking the vertices, that is if we draw points on a piece of paper, each point representing a vertex, then we would draw an arrow from vertex *i * to vertex *j * if and only if the pair (*i*, *j*) is an element of the relation *E*. Graphs can be used to represent communication networks where each person is represented by a vertex, and an edge links vertex *i* to vertex *j* if and only if person *i* communicates with person *j*.

The relation *E* can also be used to define the *adjacency matrix* of a graph. Simply define the adjacency matrix to be *A* = [aij] where aij= 1 iff (*i*, *j*) belongs to *E*, and 0 otherwise.

1. Given an arbitrary graph with edge-set *E* and adjacency matrix *A*, prove that *E* is symmetric as a relation iff *A* is symmetric as a matrix. (Hint 1: Start first with the assumption that *E* is symmetric, and deduce that *A* is symmetric. Then, assume that *A* is symmetric, and deduce that *E* is symmetric. Hint 2: This is a very easy problem. Don't make it harder than it is.)

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Here's Here's how I deal with it. Is that making any sense?

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i. Let *A* be symmetric matrix A=(aij). then aij=aji=1 which means in the graph, vertex *i* and vertex *j* are linked by E => *i*E*j* = *j*E*i*. Therefore, E is a symmetric relation.

ii. Let *E* be a symmetric relation. *i*E*j* => *j*E*i* , Which will be shown in the graph as an arrow from vertex *i* to vertex *j* and an arrow from vertex *j* to vertex *i*. => aij=aji=1 => *A* is a symmetric matrix.