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Math 170 Unit 3 Methods of Proof
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Math 170 Unit 3: Methods of Proof
A is a mathematical statement that is true and can be verified as true. theorem
A of a theorem is written verification that shows that the
theorem is definitely true.
proof
A is an exact, unambiguous explanation of the meaning of a
mathematical word or phrase
definition
• should be understandable
• should be convincing to anyone who has the requisite background and
knowledge
• This knowledge includes and understanding of the meaning of
mathematical words, phrases and symbols.
• To avoid ambiguity, the writer and the reader must agree on the exact
meanings of all the words or phrases
Math 170 Unit 3: Methods of ProofDefinitions
Math 170 Unit 3 Methods of Proof
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Math 170 Unit 3: Methods of Proof
Method 1:
Direct Proof
Example 1.1 : Prove 2If is even, then is even.n n
Math 170 Unit 3: Methods of Proof
Example 1.2 : Prove
Let , , and be integers. If and , then .a b c a b b c a c
Math 170 Unit 3 Methods of Proof
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Math 170 Unit 3: Methods of Proof
Method 2:
Proof By Contrapositive
p q q p→ ≡ →∼ ∼
If , then p q
Original conditional
If , then q p
Contrapositive
∼ ∼
Example 2.1 : Prove 2If is even, then is even.n n
From Book of Proof by Richard Hammack
Math 170 Unit 3: Methods of Proof
Example 2.2 : Prove
If 5 does not divide , then 5 does not divide and 5 does not divide .xy x y
Math 170 Unit 3 Methods of Proof
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Math 170 Unit 3: Methods of Proof
Method 3:
Proof By Contradiction
1. Assume what you are trying to prove is false.
2. Show how that leads to a contradictory
statement
3. Conclude that the original statement must have
been true.
Example 3.1 : Prove
The sum of a rational and an irrational is irrational.
From Book of Proof by Richard Hammack
Math 170 Unit 3: Methods of Proof
Example 3.2 : Prove
The number 2 is irrational.
Math 170 Unit 3 Methods of Proof
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Math 170 Unit 3: Methods of Proof
Method 4:
Proof By Induction
From Book of Proof by Richard Hammack
Math 170 Unit 3: Methods of Proof
Example 4.1 : Prove
( )1For any natural number , it follows that 1 2 3 .
2
n nn n
++ + + + =�
Math 170 Unit 3 Methods of Proof
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Math 170 Unit 3: Methods of Proof
Math 170 Unit 3: Methods of Proof
