The Proof Page

by D. A. Kouba

Section 1.4- Bacic Proof Methods I- Direct Proof, Proof by Cases, and Proof by Working Backward

In this section we will introduce specific types or methods of proof of mathematical statements. They include direct proof, proof by cases, and proof by working backward. We will use the following well known facts :

:

- 1. Integer is
*even*if for some integer . - 2. Integer is
*odd*if for some integer . - 3. Integer
*divides*integer , written , , if for some integer . NOTE : This definition applies only to integers. - 4. Natural number (positive integer) is
*prime*if its only natural number factors are and . truein - 5. The
*absolute value*of a real number , written , is defined in the following way : if ; if .

DIRECT PROOF

The

- 1. Begin with a clear written statement of the given facts or assumptions.

- 2. Next provide a clear written statement of what is to be proven.

- 3. Then write the body of the proof, a sequence of logical steps or consequences leading to the desired result. Provide clear reasoning or substantiation for each step in the proof. A good rule of thumb is this. Treat each proof that you write as a writing exercise as well as a mathematical exercise. Remember that you are trying to convince your reader of the validity of your proof through the clarity and simplicity of your organization and logical reasoning. Finish your proof with a clear statement of that which was to be proven. (Many proofs are closed with the letters QED. This refers to the Latin ``quod est demonstratum," roughly meaning ``which is proven or demonstrated.") Shortcuts in proofs are to be avoided at this stage. Writing more details is better for those of you who are just learning how to write proofs.

: Prove that if and , then .

Proof: ASSUME that and . Thus and for some integers and . SHOW that , i.e., show that for some integer . Then

(by assumptions)

(by distributive property)

,

QED

Here is another example of a direct proof.

: Prove that if is even and is odd, then is odd .

Proof: ASSUME that is even and is odd. Thus and for some integers and . SHOW that is odd, i.e., show that for some integer . Then

(by assumptions)

(by distributive property)

(adding zero)

(by commutative property)

(by distributive property)

,

QED

: Write clear and complete direct proofs for each of the following mathematical statements. Find solutions HERE : Page 1 , Page 2 .

- 1. If and , then
.

- 2. If and , then
.

- 3. If is even and is odd, then
is odd .

- 4. If is odd, then is even .

PROOF BY CASES

A

- 1. Begin with a clear written statement of the given facts or assumptions.
- 2. Next provide a clear written statement of what is to be proven.
- 3. Now determine all possible cases which must be considered in order to prove the mathematical statement.
- 4. Then write the body of the proof. For each case, this must include a sequence of logical steps or consequences leading to the desired result. Provide clear reasoning or substantiation for each step in the proof. A good rule of thumb is this. Treat each proof that you write as a writing exercise as well as a mathematical exercise. Remember that you are trying to convince your reader of the validity of your proof through the clarity and simplicity of your organization and logical reasoning. Finish your proof with a clear statement of that which was to be proven. Shortcuts in proofs are to be avoided at this stage. Writing more details is better for those of you who are just learning how to write proofs.

: Prove that if and are real numbers, where , then .

Proof: ASSUME that and are real numbers and . SHOW that . We will consider the following cases.

: Assume that and , so that . Then , , and .

: Assume that and , so that . Then , , and .

: Assume that and , so that . Then , , and .

: Assume that and , so that . Then , , and .

Thus, for all possible cases, it has been proven that .

QED

Here is another example of a proof by cases.

: Prove that for all real numbers .

Proof: ASSUME that is a real number. SHOW that . We will consider the following cases.

: Assume that . Then and , so that , i.e., .

: Assume that . Then and , so that , i.e., .

Thus, for all possible cases, it has been proven that .

QED

: Write clear and complete proofs by cases for each of the following mathematical statements. Find solutions HERE : Page 1 , Page 2 , Page 3 .

- 1. If is a real number, then
.

- 2. If is a real number, then
.

- 3. The expression is odd for all integers .

- 4. If is an even integer, then or for some integer .

- 5. If and are real numbers, then .

PROOF BY WORKING BACKWARD

A

- 1. Begin with a clear written statement of the given facts or assumptions.
- 2. Next provide a clear written statement of what is to be proven.
- 3. Then write the body of the proof. Begin with the final result, which must be PROVEN TRUE, working backward step-by-step, writing equivalent statements, until a connection is made with the assumptions of the problem or some other TRUE statement(s). Provide clear reasoning or substantiation for each step in the proof. A good rule of thumb is this. Treat each proof that you write as a writing exercise as well as a math exercise. Remember that you are trying to convince your reader of the validity of the proof through the clarity and simplicity of your organization and logical reasoning. Finish your proof with a clear statement of that which was to be proven. Shortcuts in proofs are to be avoided at this stage. Writing more details is better for those of you who are just learning how to write proofs.

: Prove that for all real numbers and .

Proof: ASSUME that and are real numbers. SHOW that . But

. . . is TRUE

. . . is TRUE

. . . is TRUE

. . . is TRUE .

QED

: Write clear and complete proofs by working backward for each of the following mathematical statements. Find solutions HERE : Page 1 , Page 2 , Page 3 , Page 4 .

- 1. The expression
for all real numbers .

- 2. If is divisible by 3 for some integer , then
is divisible by 3.

- 3. The expression
for all real numbers .

- 4. There is a fixed positive integer for which for all integers .

RETURN to The Proof Page .

Please e-mail your comments, questions, or suggestions to D. A. Kouba at
kouba@math.ucdavis.edu .