techniques in mathematical proofs

techniques in mathematical proofs

The following example (from ring theory) illustrates the one aspect of proofs in mathematics: proving the

existence

of certain mathematical .

Statement:

Let

R

be a ring such that

1

-

a

⁢

b

is right invertible, with

a

,

b

∈

R

. Then

1

-

b

⁢

a

is right invertible.

This statement will be proven here using two methods. The first method is called an

existential proof

(also known as an

existence proof

), in which one only seeks to prove that the mathematical in question exists,

not

to show how to obtain it. The second method is called a

constructive proof

, in which one actually shows how to obtain the mathematical in question.

Existential proof:

Since

1

-

a

⁢

b

∈

R

is right invertible,

(

1

-

a

⁢

b

)

⁢

R

=

R

. Now,

(

1

-

b

⁢

a

)

⁢

R

⊇

(

1

-

b

⁢

a

)

⁢

b

⁢

R

=

b

⁢

(

1

-

a

⁢

b

)

⁢

R

=

b

⁢

R

.

So

(

b

⁢

a

)

⁢

R

=

b

⁢

(

a

⁢

R

)

⊆

b

⁢

R

⊆

(

1

-

b

⁢

a

)

⁢

R

,

and consequently,

R

=

(

1

-

b

⁢

a

)

⁢

R

+

(

b

⁢

a

)

⁢

R

⊆

(

1

-

b

⁢

a

)

⁢

R

,

showing that

1

∈

(

1

-

b

⁢

a

)

⁢

R

.

Notice, we merely demonstrated the existence of a

right inverse

of

1

-

b

⁢

a

without actually finding such an . The next proof in fact finds a right inverse of

1

-

b

⁢

a

.

Constructive proof:

Since

1

-

a

⁢

b

∈

R

is right invertible, let

c

∈

R

be a right inverse so that

1

=

(

1

-

a

⁢

b

)

⁢

c

. We seek to construct a right inverse of

1

-

b

⁢

a

in terms of

a

,

b

,

and

c

. Rewriting the

equation

, we have

a

⁢

b

⁢

c

=

c

-

1

. Then,

(

1

-

b

⁢

a

)

⁢

b

⁢

c

=

b

⁢

c

-

b

⁢

a

⁢

b

⁢

c

=

b

⁢

c

-

b

⁢

(

c

-

1

)

=

b

.

We have just expressed

b

in terms of

1

-

b

⁢

a

. Next, multiply

a

on the right to each term on both sides of the equation, to get

b

⁢

a

=

(

1

-

b

⁢

a

)

⁢

b

⁢

c

⁢

a

.

Then, negate both terms and add 1, to get

1

-

b

⁢

a

=

1

-

(

1

-

b

⁢

a

)

⁢

b

⁢

c

⁢

a

.

Finally, rearranging the terms and we have

1

=

(

1

-

b

⁢

a

)

+

(

1

-

b

⁢

a

)

⁢

b

⁢

c

⁢

a

=

(

1

-

b

⁢

a

)

⁢

(

1

+

b

⁢

c

⁢

a

)

,

showing that
a right inverse of

1

-

b

⁢

a

exists by explicitly constructing one.

Many other techniques are used in proving mathematical statements. Proof by mathematical induction,

proof by contradiction

, proof by contrapositive, and proof by exhaustion are just some of the major techniques (a is in the entry “irrational to an irrational power can be

rational

”).

As this entry is still in its very rough form, PM users are welcome and encouraged to refine and provide additional techniques with interesting and illustrative examples!

Title

techniques in mathematical proofs

Canonical

name

TechniquesInMathematicalProofs

Date of creation

2013-03-22 14:46:15

Last modified on

2013-03-22 14:46:15

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

18

Author

CWoo (3771)

Entry type

Feature

Classification

msc 00A35

Classification

msc 03F07

Related topic

IrrationalToAnIrrationalPowerCanBeRational

Related topic

ExistentialTheorem

Related topic

IABIsInvertibleIfAndOnlyIfIBAIsInvertible

Defines

existential proof

Defines

existence proof

Defines

constructive proof

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