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Brief Review of Proof Techniques

Brief Review of Proof Techniques

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Brief Review of Proof Techniques

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  1. Brief Review of Proof Techniques

  2. 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).

  3. Three general methods Proof by a direct argument. Proof by contradiction. Proof by induction.

  4. 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.

  5. 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.

  6. 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. □

  7. 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.

  8. 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. □

  9. 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.

  10. 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. 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.

  12. 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. □

  13. 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.

  14. 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. □

  15. 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. 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 □