The Forward-Backward Method

1
The Forward-Backward Method

• The First Method To Prove
• If A, Then B.

2
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 cant be asked or
  cant 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

3
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.
4
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.

5
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 ?

6
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.

7
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.

8
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.

9
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

10
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

11
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 ½baseheight 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

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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. ?

13
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. ?

14
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
  (x2y2) 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. ?

15
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. ?

16
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.

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Recognize as If A, then B.

• If fX?X and gX?X are both one-to-one functions,
  then f o g is one-to-one.
• A The functions fX?X and gX?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?

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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?

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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 fX?X and gX?X are both
• one-to-one functions and if f o g ( x )
  f o g ( y ),
• then x y.

20

• So we begin with B x y and note that,
  since we dont know anything about x y except
  that x y are in the domain X, we cant pose a
  reasonable key question for B so we should begin
  the Forward Process for this new if-then
  statement.

21
The Forward Process

• A The functions fX?X and gX?X are both
• one-to-one functions and f o g ( x )
  f o g ( y )
• A1 f(g(x)) f(g(y)) (definition of
  composition)
• A2 g(x) g(y) (f is one-one)
• A3 x y (g is one-one)
• Note that A3 is B so we have proved the
  statement
• Now write the proof.

22
Write the Proof

• Statement Reason
• A The functions fX?X Given
• and gX?X are both
• one-to-one.
• A f o g ( x ) f o g ( y ) Assumed to
  prove f o g is 1-1
• A1 f(g(x)) f(g(y)) definition of
  composition
• A2 g(x) g(y) f is 1-1 by A
• A3 x y g is 1-1 by A
• B f o g is 1-1 definition of 1-1

23
Condensed Proof

• Suppose the fX?X and gX?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. ?