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## The Forward-Backward Method

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The Forward-Backward Method General Outline (Simplified)

- Recognize the statement “If A, then B.”
- Use the Backward Method repeatedly until A is reached or the “Key Question” can’t be asked or can’t be answered.
- Use the Forward Method until the last statement derived from the Backward Method is obtained.
- Write the proof by
- starting with A, then
- those statements derived by the Forward Method, and then
- those statements (in opposite order) derived by the Backward Method

An Example:

If the right triangle XYZ with sides of lengths x and y, and hypotenuse of length z, has an area of z2/4, then the triangle XYZ is isosceles.

- Recognize the statement “If A, then B.”

A: The right triangle XYZ with sides of lengths x and y, and hypotenuse of length z, has an area of z2/4.

B: The triangle XYZ is isosceles.

The Backward Process

- Ask the key question:

“How can I conclude that statement B is true?”

- must be asked in an ABSTRACT way
- must be able to answer the key question
- there may be more than one key question
- use intuition, insight, creativity, experience, diagrams, etc.
- let statement A guide your choice
- remember options - you may need to try them later
- Answer the key question.
- Apply the answer to the specific problem
- this new statement B1 becomes the new goal to prove from statement A.

The Backward Process: An Example

- Ask the key question: ‘How can I conclude that statement :

“The triangle XYZ is isosceles” is true?’

- ABSTRACT key question:

“ How can I show that a triangle is isosceles?”

- Answer the key question.
- Possible answers: Which one? ... Look at A: The right triangle XYZ with sides of lengths x and y, and hypotenuse of length z, has an area of z2/4
- Show the triangle is equilateral.
- Show two angles of the triangle are equal.
- Show two sides of the triangle are equal.
- Apply the answer to the specific problem
- New conclusion to prove is B1: x = y.
- Why not x = z or y = z ?

Backward Process Again:

- Ask the key question: ‘How can I conclude that statement :

“B1: x = y” is true?’

- ABSTRACT key question:

“ How can I show two real numbers are equal?”

- Answer the key question.
- Possible answers: Which one? ... Look at A.
- Show each is less than and equal to the other.
- Show their difference is 0.
- Apply the answer to the specific problem
- New conclusion to prove is B2: x - y = 0.

Backward Process Again:

- Ask the key question: ‘How can I conclude that statement :

“B2: x - y = 0” is true?’

- ABSTRACT key question:

No reasonable way to ask a key question. So,

Time to use the Forward Process.

The Forward Process

- From statement A, derive a conclusion A1.
- Let the last statement from the Backward Process guide you.
- A1 must be a logical consequence of A.
- If A1 is the last statement from the Backward Process then the proof is complete,
- Otherwise use statements A and A1 to derive a conclusion A2.
- Continue deriving A3, A4, .. until last statement from the Backward Process is derived.

Variations of the Forward Process

- A derivation might suggest a way to ask or answer the last key question from the Backward Process; continuing the Backward Process.
- An alternative question or answer may be made for one of the steps in the Backward Process; continuing the Backward Process from that point on.
- The Forward-Backward Method might be abandoned for one of the other proof methods

The Forward Process: Continuing the Example

- Derive from statement A: The right triangle XYZ with sides of lengths x and y, and hypotenuse of length z, has an area of z2/4.
- A1: ½ xy = z2/4 (the area = the area)
- A2: x2 + y2 = z2 ( Pythagorean theorem)
- A3: ½ xy = (x2 + y2)/4 ( Substitution using A2 and A1)
- A4: x2 -2xy + y2 = 0 ( Multiply A3 by 4; subtract 2xy )
- A5: (x -y)2 = 0 ( Factor A4 )
- A6: (x -y) = 0 ( Take square root of A5)
- Note: A6 B2, so we have found a proof

Write the Proof

Statement Reason

- A: The right triangle XYZ with sides of lengths x and y, and hypotenuse of length z, has an area of z2/4.

Given

- A1: ½ xy = z2/4 Area = ½base*height; and A
- A2: x2 + y2 = z2 Pythagorean theorem
- A3: ½ xy = (x2 + y2)/4 Substitution using A2 and A1
- A4: x2 -2xy + y2 = 0 Multiply A3 by 4; subtract 2xy
- A5: (x -y)2 = 0 Factor A4
- B2: (x -y) = 0 Take square root of A5
- B1: x = y Add y to B2
- B: XYZ is isosceles B1 and definition of isosceles

Write Condensed Proof - Forward Version

From the hypothesis and the formula for the area of a right triangle, the area of XYZ = ½ xy = ¼ z2. By the Pythagorean theorem, (x2 + y2)= z2, and on substituting (x2 + y2) for z2 and performing some algebraic manipulations one obtains (x -y)2 = 0. Hence x = y and the triangle XYZ is isosceles.

Write Condensed Proof - Forward & Backward Version

The statement will be proved by establishing that x = y, which in turn is done by showing that (x -y)2 = (x2 -2xy + y2) = 0. But the area of the triangle is ½ xy = ¼ z2, so that 2xy = z2. By the Pythagorean theorem, x2 + y2 = z2 and hence (x2 + y2)= 2xy, or (x2 -2xy + y2 ) = 0.

Write Condensed Proof - Backward Version

To reach the conclusion, it will be shown that x = y by verifying that (x -y)2 = (x2 -2xy + y2) = 0, or equivalently, that (x2 + y2)= 2xy. This can be established by showing that 2xy = z2, for the Pythagorean theorem states that (x2+y2) = z2. In order to see that 2xy = z2, or equivalently, that ½ xy = ¼ z2, note that ½ xy is the area of the triangle and it is equal to ¼ z2 by hypothesis, thus completing the proof.

Write Condensed Proof - Text Book or Research Version

The hypothesis together with the Pythagorean theorem yield (x2 + y2)= 2xy; hence (x -y)2 = 0. Thus the triangle is isosceles as required.

Another Forward-Backward Proof

Prove: The composition of two one-to-one functions is one-to-one.

- Recognize the statement as “If A, then B.”

Recognize as “If A, then B.”

- If f:XX and g:XX are both one-to-one functions, then f o g is one-to-one.
- A: The functions f:XX and g:XX are both one-to-one.
- B: The function f o g: XX is one-to-one.
- What is the key question and its answer?

The Key Question and Answer

- Abstract question

How do you show a function is one-to-one.

- Answer: Assume that if the functional value of two arbitrary input values x and y are equal then x = y.
- Specific answer -

B1: If f o g ( x ) = f o g ( y ), then x = y.

- How do you show B1? What is the key question?

The Key Question and Answer

- How do you show

B1: If f o g ( x ) = f o g ( y ), then x = y.

- Answer:

We note that B1 is of the form If A`, the B`, and use the Forward-Backward method to prove the statement

If A and A`, then B`. ie.,

If the functions f:XX and g:XX are both

one-to-one functions and if f o g ( x ) = f o g ( y ),

then x = y.

So we begin with B` : x = y and note that, since we don’t know anything about x & y except that x & y are in the domain X, we can’t pose a reasonable key question for B` so we should begin the Forward Process for this new if-then statement.

The Forward Process

- A`: The functions f:XX and g:XX are both

one-to-one functions and f o g ( x ) = f o g ( y )

- A`1: f(g(x)) = f(g(y)) (definition of composition)
- A`2: g(x) = g(y) (f is one-one)
- A`3: x = y (g is one-one)

Note that A`3 is B` so we have proved the statement

Now write the proof.

Write the Proof

Statement Reason

- A: The functions f:XX Given

and g:XX are both

one-to-one.

- A`: f o g ( x ) = f o g ( y ) Assumed to prove f o g is 1-1
- A`1: f(g(x)) = f(g(y)) definition of composition
- A`2: g(x) = g(y) f is 1-1 by A
- A`3: x = y g is 1-1 by A
- B: f o g is 1-1 definition of 1-1

Condensed Proof

Suppose the f:XX and g:XX are both one-to-one.

To show f o g is one-to-one we assume f o g ( x ) = f o g ( y ).

Thus f(g(x)) = f(g(y) and since f is one-to-one, g(x) = g(y).

Since g is also one-to-one x = y.

Therefore f o g is one-to-one.

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