## Homework 5, Due Thursday, 03/03

For this homework assignment, you are asked to provide 3 proofs. Remember that proofs are written in English. You proof should not be a sequence of arithmetic equations. There must be a narrative composed of complete English sentences, correctly punctuated, with math symbols mixed in as appropriate, which convinces the reader that the claim is correct.

1. Existence Proof. Let G be an undirected graph that is connected and does not have any cycles. Prove that there exists a vertex v in G that has a degree of 1. (I.e., show that there must be a vertex that is attached to only one edge.)

We need to make the additional assumption that the graph G has at least two vertices. Otherwise a graph with a single vertex and zero edges is technically connected and has no cycles but does not have any vertices with degree 1.

2. Uniqueness Proof. [From Epp 4/e]
Prove that there is a unique prime number of the form n2 + 2n − 3, where n is a positive integer.
Hint: try some numbers.
Hint, hint: factor.

3. Equivalence Proof. [Adapted from Rosen 5/e]
See write up on proving set equality for an example format.

Prove the equality of the two sets below by showing that every element of the set on the left hand side of the equality is also an element of the set on the right hand side, and vice versa. (I.e., do not prove this using algebraic identities.)

(BA) ∪ (CA)  =  (BC) − A