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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.

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The Forward-Backward Method


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the forward backward method
The Forward-Backward Method

The First Method To Prove

If A, Then B.

the forward backward method general outline simplified
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
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
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
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
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 again7
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
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
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
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
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
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
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
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
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
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
Recognize as “If A, then B.”
  • If f:XX and g:XX are both one-to-one functions, then f o g is one-to-one.
  • A: The functions f:XX and g:XX are both one-to-one.
  • B: The function f o g: XX is one-to-one.
  • What is the key question and its answer?
the key question and 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 answer19
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:XX and g:XX are both

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

then x = y.

slide20
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 process21
The Forward Process
  • A`: The functions f:XX and g:XX 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 proof22
Write the Proof

Statement Reason

  • A: The functions f:XX Given

and g:XX 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
Condensed Proof

Suppose the f:XX and g:XX 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. 