The Forward-Backward Method

The First Method To Prove

If A, Then B.

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.

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.

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

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.