Here's the LONG question...

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Agraphis a finite set ofverticesValong with a relationEonV. The elements of the relationEare callededges(and the vertices are sometimes callednodes). 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 vertexito vertexjif and only if the pair (i,j) is an element of the relationE. Graphs can be used to represent communication networks where each person is represented by a vertex, and an edge links vertexito vertexjif and only if personicommunicates with personj.

The relationEcan also be used to define theadjacency matrixof a graph. Simply define the adjacency matrix to beA= [aij] where aij= 1 iff (i,j) belongs toE, and 0 otherwise.

1. Given an arbitrary graph with edge-setEand adjacency matrixA, prove thatEis symmetric as a relation iffAis symmetric as a matrix. (Hint 1: Start first with the assumption thatEis symmetric, and deduce thatAis symmetric. Then, assume thatAis symmetric, and deduce thatEis 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. LetAbe symmetric matrix A=(aij). then aij=aji=1 which means in the graph, vertexiand vertexjare linked by E =>iEj=jEi. Therefore, E is a symmetric relation.

ii. LetEbe a symmetric relation.iEj=>jEi, Which will be shown in the graph as an arrow from vertexito vertexjand an arrow from vertexjto vertexi. => aij=aji=1 =>Ais a symmetric matrix.