(12 points) Recall that in class, we proved that 2 was irrational. In this problem, we will explore a differe proof that 2 is irrational and you will fill in the details. proof Suppose by way of contradiction that there exist integers p and q>0 such that 2=p/q. (a) Explain why this statement is a valid way to start the proof. Define the set A={nZ+a=2aZ}. (b) Justify that AN. (c) Justify that A=. Now, we can use the Well-Ordering principle on A. (d) What can we conclude about A using the well-ordering principle? Based on our assumption, 2=p/q. Square both sides and multiply by q2 to get p2=2q2. Subtract pq from both sides to get p2pq=2q2pq. This can be factored as: p(pq)=q(2qp) and can be rearranged as: qp=pq2qp (e) This last rearrangement relies on dividing by q and by pq. Give a justification about why q=0 and why pq=0. Now we have that 2=qpand2=pq2qp (f) Finish the proof by finding a contradiction using the well-ordering principle