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I've been trying to prove things more often because I haven't done it a lot and I'm interested in a mathy career. I started reading Sipser's Introduction to the Theory of Computation and came across a chance to try and prove the statement 'For every graph G, the sum of the degrees of all nodes in G is even.' I couldn't find other proofs online, so I thought I'd share mine here before I look at the book, especially because mine might be completely different and I wouldn't really know if it was any good.
A graph G equals the set of the set of nodes/vertices V and the set of edges E. That is, G = {V, E}.
Let G be the empty graph with no nodes and no edges. The sum of the degrees of the nodes of this graph is zero, which is even.
Let G be the graph with one node and no edges. The sum of the degrees of the nodes of this graph is zero, which is even.
Let G be an arbitrary, non-empty graph such that the sum of the degrees of the nodes in G is even.
Let G' be a graph identical to G in all respects except that it contains an additional node that is a member of an additional pair in E with one other node. (That is, 'make a new node' and 'make an edge' to attach it to an existing node with.) The degree of a node equals the number of pairs in E of which the node is a member. Each pair contains two elements, so that if a graph G has i edges and j equals the sum of the degrees of all nodes in G, then the sum of the degrees of all nodes in a graph G' with i+1 edges will equal j+2. Because this is true for an arbitrary, non-empty graph G, it is true for every non-empty graph G. j is even by assumption, and the sum of two even numbers is even, so j+2 is even. Because this is true for an arbitrary, non-empty graph G', it is true for every non-empty graph G.
For every non-empty graph G, the sum of the degrees of all nodes in G is even. The sum of the degrees of all nodes in the empty graph is even. Therefore, for every graph G, the sum of the degrees of all nodes in G is even.
FYI, this is called the sum of degrees theorem. In fact, the sum of degrees is not only an even number, but twice the number of edges in the graph. This is due to Euler, I think. He used the famous Koenigsberg bridges problem as a motivation for thinking about graphs.
Good work on thinking about proofs, +1 to you.