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” 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

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 = 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

12. 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 (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. 

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.”

17. 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?

18. 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?

19. 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.

20. 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.

21. 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.

22. 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

23. 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. 