Brief Review of Proof Techniques Frequently, presenters must deliver material of a technical nature to an audience unfamiliar with the topic or vocabulary. The material may be complex or heavy with detail. To present technical material effectively, use the following guidelines from Dale Carnegie Training®.   Consider the amount of time available and prepare to organize your material. Narrow your topic. Divide your presentation into clear segments. Follow a logical progression. Maintain your focus throughout. Close the presentation with a summary, repetition of the key steps, or a logical conclusion. Keep your audience in mind at all times. For example, be sure data is clear and information is relevant. Keep the level of detail and vocabulary appropriate for the audience. Use visuals to support key points or steps. Keep alert to the needs of your listeners, and you will have a more receptive audience. 11/27/2018

What is a proof? A theorem is a proven mathematical statement. A proof is a sequence of statements that form an argument (to prove sth, say a theorem). 11/27/2018

Three general methods Proof by a direct argument. Proof by contradiction. Proof by induction. 11/27/2018

Three general methods Proof by a direct argument. With this method, you try to find the core of the proof and then use a direct mathematical argument. 11/27/2018

Three general methods Proof by a direct argument. With this method, you try to find the core of the proof and then use a direct mathematical argument. Example. Theorem. Σvdeg(v) is even. 11/27/2018

Three general methods Proof by a direct argument. With this method, you try to find the core of the proof and then use a direct mathematical argument. Example. Theorem. Σvdeg(v) is even. Proof. Let e=<u,v> be any edge in the graph. When counting deg(u) and deg(v), e is counted once each time. Therefore, each edge e contributes 2 to Σvdeg(v). Therefore, Σvdeg(v)=2|E|, which is always even. □ 11/27/2018

Three general methods Proof by a direct argument. With this method, you try to find the core of the proof and then use a direct mathematical argument. Example. Theorem. A tree with n nodes has n-1 edges. 11/27/2018

Three general methods Proof by a direct argument. With this method, you try to find the core of the proof and then use a direct mathematical argument. Example. Theorem. A tree T with n nodes has n-1 edges. Proof. Let T=Tn. We know that each tree has at least 2 leaves. So we delete a leave node as well as the edge incident to it to obtain Tn-1, which is again a tree. We can repeat this process n-2 times (delete one node for one edge) until we have T2, which has only one edge. Clearly, Tn has (n-2)+1=n-1 edges. □ 11/27/2018

Three general methods Proof by contradiction. With this method, you assume that the statement you want to prove is not true. Then you try to obtain a contradiction (either with the definition or some known facts). Example. Theorem. A tree T with n nodes has n-1 edges. 11/27/2018

Three general methods Proof by contradiction. With this method, you assume that the statement you want to prove is not true. Then you try to obtain a contradiction (either with the definition or some known facts). Example. Theorem. A tree T with n nodes has n-1 edges. Proof. Assume that T doesn’t have n-1 edges. So it can contain either >n-1 edges or <n-1 edges. If it has more than n-1 edges, then T must contain a cycle. If it has less than n-1 edges, then T is disconnected. In either cases, we have a contradiction as by definition T is a connected acyclic graph. □ 11/27/2018

Three general methods Proof by contradiction. With this method, you assume that the statement you want to prove is not true. Then you try to obtain a contradiction (either with the definition or some known facts). Example. Theorem. √2 is irrational. 11/27/2018

Three general methods Proof by contradiction. With this method, you assume that the statement you want to prove is not true. Then you try to obtain a contradiction (either with the definition or some known facts). Example. Theorem. √2 is irrational. Proof. Assume that √2 is rational. By definition √2=m/n, with m,n being integers and gcd(m,n)=1. Square both sides of √2=m/n, we have 2n2=m2 . Then m must be even. Let m=2k. We have 2n2=(2k)2=4k2. Then n2=2k2, so n is even as well. Then gcd(m,n)≠1. A contradiction. □ 11/27/2018

Three general methods Proof by induction. With this method, you have to check the Basis, then make an assumption that the claim (to be proven) is true up to certain k and finally you should that the claim is still true for k+1. This method usually can only be used to prove claims related to natural numbers. Example. Theorem. A tree T with n nodes has n-1 edges. 11/27/2018

Three general methods Proof by induction. With this method, you have to check the Basis, then make an assumption that the claim (to be proven) is true up to certain k and finally you should that the claim is still true for k+1. This method usually can only be used to prove claims related to natural numbers. Example. Theorem. A tree T with n nodes has n-1 edges. Proof. Basis. When n=1, T has n-1=0 edges. So the claim is correct. Inductive Hypothesis. Assume that the claim is true for all T with k vertices. Inductive Step. Given a tree with k+1 vertices, TK+1, we know that every tree has at least two leaves. So by pruning a leave node and its incident edge e from Tk+1, we obtain another tree T’ with k vertices. By IH, T’ has k-1 edges. Adding the edge e back, Tk+1 has (k - 1)+1 = k = (k+1) – 1 edges. □ 11/27/2018

Three general methods Proof by induction. =(k+1)2[(k+1)+1]2/4 □ With this method, you have to check the Basis, then make an assumption that the claim (to be proven) is true up to certain k and finally you should that the claim is still true for k+1. This method usually can only be used to prove claims related to natural numbers. Example. Theorem. 13+23+…+n3=n2(n+1)2/4. Proof. Basis. When n=1, 13=12(1+1)2/4=1. Inductive Hypothesis. Assume that 13+23+…+k3=k2(k+1)2/4. Inductive Step. 13+23+…+k3+(k+1)3=k2(k+1)2/4 + (k+1)3 = [k2/4+(k+1)](k+1)2 = (k+1)2(k+2)2/4 =(k+1)2[(k+1)+1]2/4 □ 11/27/2018