Download pdf - Math 170 Unit 3 Methods of Proof - University of Pennsylvaniahans.math.upenn.edu/~rimmer/math170/notes/unit3methodsofproof.pdf · From Book of Proof by Richard Hammack Math 170 Unit

Math 170 Unit 3 Methods of Proof

1

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


2


Method 1:

Direct Proof

Example 1.1 : Prove 2If is even, then is even.n n


Example 1.2 : Prove

Let , , and be integers. If and , then .a b c a b b c a c


3


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


Example 2.2 : Prove

If 5 does not divide , then 5 does not divide and 5 does not divide .xy x y


4


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.



Example 3.2 : Prove

The number 2 is irrational.


5


Method 4:

Proof By Induction



Example 4.1 : Prove

( )1For any natural number , it follows that 1 2 3 .

2

n nn n

++ + + + =�


6

