Prove the following statement about graph theory: If a new edge (but no new vertex) is added to a tree,

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Prove the following statements: 

1. (8 points) V7 is irrational. 2. (8 points) The sum of an odd number of odd numbers must be odd. 3. (8 points) Let f : R —> HI and g : R be two real-valued functions. Let S be the solution set of the equation f (x) = g (x) , and let T be the solution set of the modified equation (f (x))2 = (g (x))2 . Show that S is a subset of T , but that S and T are not necessarily equal. 

Extra Credit (+3 points): 

Prove the following statement about graph theory: If a new edge (but no new vertex) is added to a tree, then the new graph will contain a circuit.

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