Brief Review of Proof Techniques

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Brief Review of Proof Techniques

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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).
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Three general methods
Proof by a direct argument. Proof by
contradiction. Proof by induction.
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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.
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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.
Svdeg(v) is even.
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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.
Svdeg(v) is even. Proof. Let eltu,vgt 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 Svdeg(v). Therefore,
Svdeg(v)2E, which is always even. ?

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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.
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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
TTn. 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)1n-1 edges. ?
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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.
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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 doesnt have
n-1 edges. So it can contain either gtn-1 edges or
ltn-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.
?
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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. v2 is irrational.
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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. v2 is
irrational. Proof. Assume that v2 is rational. By
definition v2m/n, with m,n being integers and
gcd(m,n)1. Square both sides of v2m/n, we have
2n2m2 . Then m must be even. Let m2k. We have
2n2(2k)24k2. Then n22k2, so n is even
as well. Then gcd(m,n)?1. A contradiction.
?
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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 k1. 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.
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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 k1. 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 n1, T has n-10 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 k1
vertices, TK1, we know that every tree has at
least two leaves. So by pruning a leave node and
its incident edge e from Tk1, we obtain another
tree T with k vertices. By IH, T has k-1 edges.
Adding the edge e back, Tk1 has (k - 1)1 k
(k1) 1 edges. ?
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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 k1. This method usually can only be used to
prove claims related to natural numbers. Example.
Theorem. 1323n3n2(n1)2/4. Proof. Basis.
When n1, 1312(11)2/41. Inductive Hypothesis.
Assume that 1323k3k2(k1)2/4. Inductive
Step. 1323k3(k1)3k2(k1)2/4 (k1)3
k2/4(k1)(k1)2 (k1)2(k2)2/4 (k1)2(k1)
12/4 ?