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MAT 2720Discrete Mathematics

- Section 2.2
- More Methods of ProofPart 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

Proof by Contradiction

- Proof by Contrapositive
- Proof by Contradiction

Example 2

Analysis Proof

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.

Proof by Contradiction

- The method also work with statements other then

If P then Q

Example 3

Analysis Proof

Proof by Cases

Example 4

Analysis Proof

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.

Existence Proofs

Example 5

Analysis Proof

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.

MAT 2720Discrete Mathematics

- Section 2.4
- Mathematical Induction

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Preview

- Review Mathematical Induction
- Why?
- How?

The Needs

- Theorems involve infinitely many, yet countable,

number of statements.

Principle of Mathematical Induction (PMI)

- PMI It suffices to show
- 1. P(1) is true.
- 2. If P(k) is true, then P(k1) is also true, for

all k.

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(k1) is also true, for

all k (Inductive Step)

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.

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.

Example 1

- Use mathematical induction to prove that
- whenever n is a nonnegative integer.

Checklist A

0,

Checklist B

Checklist C

Checklist C

Checklist D

Proof by Mathematical Induction

Declare 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 AB 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 (k1)-th case. This is what you need to prove to be true. For AB 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(k1) is true. Make the formal conclusion by quoting the PMI

Example 2

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Example 3

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