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MAT 2720Discrete Mathematics
Section 2.2More Methods of Proof
Part II
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Goals Indirect Proofs
• Contrapositive• Contradiction• Proof by Contrapositive is considered as a
special case of proof by contradiction• Proof by cases• Existence proofs
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Proof by Contradiction Proof by Contrapositive
Proof by Contradiction
If then P Q
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Example 2
Analysis Proof
If is odd, then 3 2 is also odd.n n
Suppose 3 2 is even.n
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Proof by Contradiction
Analysis Proof by Contradiction of If-then Theorem• Suppose the negation of the conclusion is true. • Find a contradiction.• State the conclusion.
If is odd, then 3 2 is also odd.n n
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Proof by Contradiction The method also work with statements other
then If P then Q
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Example 3
Analysis Proof
2 is an irrational number.
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Proof by Cases
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Example 4
Analysis Proof
, ,xy x y x y
if 0if 0
x xx
x x
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Proof by Cases
Analysis Proof by Cases of If-then Theorem• Split the domain of interest into cases.• Prove each case separately.• State the conclusion.
•Note that the cases do not have to be mutually exclusive. They just have to cover all elements in the domain.
, ,xy x y x y
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Existence Proofs
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Example 5
Analysis Proof
Suppose , such that .Show that such that .
a b a bx a x b
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Existence Proofs
Analysis Existence Proof• Prove the statement by exhibiting an element in the domain of interest that satisfies the given conditions.•State the conclusion.
Suppose , such that .Show that such that .
a b a bx a x b
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MAT 2720Discrete Mathematics
Section 2.4 Mathematical Induction
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Preview Review Mathematical Induction
• Why?• How?
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The Needs…
( 1)1 2 ; 2
n n n Nn
Theorems involve infinitely many, yet countable, number of statements.
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Principle of Mathematical Induction (PMI)
PMI: It suffices to show 1. P(1) is true.2. If P(k) is true, then P(k+1) is also
true, for all k.
(1) (2) (3) (66) (67) (68)P P P P P P
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Principle of Mathematical Induction (PMI)
PMI: It suffices to show 1. P(1) is true. (Basic Step)2. If P(k) is true, then P(k+1) is also
true, for all k (Inductive Step)
(1) (2) (3) (66) (67) (68)P P P P P P
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Format of Solutions In this course, it is extremely important
for you to follow the exact solution format of using mathematical induction.
Do not skip steps.
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Format of Solutions In this course, it is extremely important
for you to follow the exact solution format of using mathematical induction.
Do not skip steps.
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Example 1Use mathematical induction to prove that
whenever n is a nonnegative integer.
2 2 2 2 ( 1)(2 1)(2 3)1 3 5 (2 1)3
n n nn
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Checklist A
2 2 2 2 ( 1)(2 1)(2 3)1 3 5 (2 1)3
n n nn
0,
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Checklist B
2 2 2 2 ( 1)(2 1)(2 3)1 3 5 (2 1)3
n n nn
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Checklist C
2 2 2 2 ( 1)(2 1)(2 3)1 3 5 (2 1)3
n n nn
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Checklist C
2 2 2 2 ( 1)(2 1)(2 3)1 3 5 (2 1)3
n n nn
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Checklist D
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Proof by Mathematical InductionDeclare P(n) and the domain of n.• Basic Step
• Write down the statement of the first case. • Do not do any simplifications or algebra on the statement of the first case.• Explain why it is true.
• For A=B type, simplify and/ or manipulate each side and see that they are the same.
• Inductive Step• Write down the k-th case. This is the inductive hypothesis.• Write down the (k+1)-th case. This is what you need to prove to be true.• For A=B type, we usually start form one side of the equation and show that it equals to the other side.
• In the process, you need to use the inductive hypothesis.• Conclude that p(k+1) is true.
• Make the formal conclusion by quoting the PMI
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Example 2
1
Show that for 1, 2,3,...
! 2nn
n
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Example 3
1
0
Show that for real number 1,
1 , for
1
nni
i
r
a rar n
r
N
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