Transcript The Forward-Backward Method

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.
• 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
 A:
Reason
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: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 oneto-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
 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: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.
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: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 Proof
Statement
 A:
The functions f:XX
and g:XX are both
one-to-one.
 A`: f o g ( x ) = f o g ( y )
 A`1: f(g(x)) = f(g(y))
 A`2: g(x) = g(y)
 A`3: x = y
 B:
f o g is 1-1
Reason
Given
Assumed to prove f o g is 1-1
definition of composition
f is 1-1 by A
g is 1-1 by A
definition of 1-1
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. 