The Forward-In reverse Technique.


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The Forward-Backward Method General Outline (Simplified). Perceive the announcement
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Slide 1

The Forward-Backward Method The First Method To Prove If A, Then B.

Slide 2

The Forward-Backward Method General Outline (Simplified) Recognize the announcement "If A, then B." Use the Backward Method more than once until An is come to or the "Key Question" can\'t be asked or can\'t be replied. Utilize the Forward Method until the last explanation got from the Backward Method is acquired. Compose the confirmation by beginning with A, then those announcements inferred by the Forward Method, and after that those announcements (in inverse request) determined by the Backward Method

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An Example: If the right triangle XYZ with sides of lengths x and y, and hypotenuse of length z, has a territory of z 2/4, then the triangle XYZ is isosceles. Perceive the announcement "If A, then B." A: The right triangle XYZ with sides of lengths x and y, and hypotenuse of length z, has a territory of z 2/4. B: The triangle XYZ is isosceles.

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The Backward Process Ask the key inquiry: "In what manner would I be able to infer that announcement B is valid?" must be asked in an ABSTRACT way should have the capacity to answer the key inquiry there might be more than one key inquiry use instinct, knowledge, inventiveness, experience, outlines, and so forth let articulation An aide your decision recollect choices - you may need to attempt them later Answer the key inquiry. Apply the response to the particular issue this new explanation B1 turns into the new objective to demonstrate from proclamation A.

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The Backward Process: An Example Ask the key inquiry: \'By what method would I be able to infer that announcement : "The triangle XYZ is isosceles" is valid?\' ABSTRACT key inquiry: " How would I be able to demonstrate that a triangle is isosceles?" Answer the key inquiry. Conceivable answers: Which one? ... Take a gander at A: The right triangle XYZ with sides of lengths x and y, and hypotenuse of length z, has a region of z 2/4 Show the triangle is equilateral. Show two points of the triangle are equivalent. Show two sides of the triangle are equivalent. Apply the response to the particular issue New conclusion to demonstrate is B1: x = y. Why not x = z or y = z ?

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Backward Process Again: Ask the key inquiry: \'By what method would I be able to infer that announcement : "B1: x = y" is valid?\' ABSTRACT key inquiry: " How would I be able to show two genuine numbers are equivalent?" Answer the key inquiry. Conceivable answers: Which one? ... Take a gander at A. Demonstrate each is not exactly and equivalent to the next. Demonstrate their distinction is 0. Apply the response to the particular issue New conclusion to demonstrate is B2: x - y = 0.

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Backward Process Again: Ask the key inquiry: \'In what manner would I be able to presume that announcement : "B2: x - y = 0" is valid?\' ABSTRACT key question: No sensible approach to ask a key inquiry. Along these lines, Time to utilize the Forward Process .

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The Forward Process From proclamation An, infer a conclusion A1. Give the last explanation from the Backward Process a chance to guide you. A1 must be a legitimate outcome of A. On the off chance that A1 is the last explanation from the Backward Process then the evidence is finished, Otherwise utilize articulations An and A1 to determine a conclusion A2. Keep determining A3, A4, .. until last proclamation from the Backward Process is inferred.

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Variations of the Forward Process An induction may propose an approach to ask or answer the last key inquiry from the Backward Process; proceeding with the Backward Process. An option question or answer might be made for one of the means in the Backward Process; proceeding with the Backward Process starting there on. The Forward-Backward Method may be surrendered for one of the other evidence techniques

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The Forward Process: Continuing the Example Derive from explanation A: The right triangle XYZ with sides of lengths x and y, and hypotenuse of length z, has a territory of z 2/4. A1: ½ xy = z 2/4 (the range = the region) A2: x 2 + y 2 = z 2 ( Pythagorean hypothesis) A3: ½ xy = (x 2 + y 2 )/4 ( Substitution utilizing A2 and A1) A4: x 2 - 2xy + y 2 = 0 ( Multiply A3 by 4; subtract 2xy ) A5: (x - y) 2 = 0 ( Factor A4 ) A6: (x - y) = 0 ( Take square base of A5) Note: A6  B2, so we have found a proof

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Write the Proof Statement Reason A: The right triangle XYZ with sides of lengths x and y, and hypotenuse of length z, has a territory of z 2/4. Given A1: ½ xy = z 2/4 Area = ½base*height; and An A2: x 2 + y 2 = z 2 Pythagorean hypothesis A3: ½ xy = (x 2 + y 2 )/4 Substitution utilizing A2 and A1 A4: x 2 - 2xy + y 2 = 0 Multiply A3 by 4; subtract 2xy A5: (x - y) 2 = 0 Factor A4 B2: (x - y) = 0 Take square foundation of A5 B1: x = y Add y to B2 B: XYZ is isosceles B1 and meaning of isosceles

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Write Condensed Proof - Forward Version From the theory and the equation for the range of a right triangle, the region of XYZ = ½ xy = ¼ z 2 . By the Pythagorean hypothesis, (x 2 + y 2 ) = z 2 , and on substituting (x 2 + y 2 ) for z 2 and playing out some mathematical controls one gets (x - y) 2 = 0. Thus x = y and the triangle XYZ is isosceles. 

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Write Condensed Proof - Forward & Backward Version The announcement will be demonstrated by building up that x = y, which thus is finished by demonstrating that (x - y) 2 = (x 2 - 2xy + y 2) = 0. Be that as it may, the zone of the triangle is ½ xy = ¼ z 2 , so that 2xy = z 2 . By the Pythagorean hypothesis, x 2 + y 2 = z 2 and subsequently (x 2 + y 2 ) = 2xy, or (x 2 - 2xy + y 2 ) = 0. 

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Write Condensed Proof - Backward Version To achieve the conclusion, it will be demonstrated that x = y by checking that (x - y) 2 = (x 2 - 2xy + y 2) = 0, or equally, that (x 2 + y 2 ) = 2xy. This can be built up by demonstrating that 2xy = z 2 , for the Pythagorean hypothesis expresses that (x 2 +y 2 ) = z 2 . With a specific end goal to see that 2xy = z 2 , or proportionately, that ½ xy = ¼ z 2 , note that ½ xy is the region of the triangle and it is equivalent to ¼ z 2 by speculation, in this way finishing the evidence. 

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Write Condensed Proof - Text Book or Research Version The theory together with the Pythagorean hypothesis yield (x 2 + y 2 ) = 2xy; subsequently (x - y) 2 = 0. In this manner the triangle is isosceles as required. 

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Another Forward-Backward Proof Prove: The structure of two balanced capacities is coordinated. Perceive the announcement as "If A, then B."

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Recognize as "If A, then B." If f:X  X and g:X  X are both coordinated capacities, then f o g is balanced. A: The capacities f:X  X and g:X  X are both balanced. B: The capacity f o g: X  X is balanced. What is the key inquiry and its answer?

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The Key Question and Answer Abstract question How would you demonstrate a capacity is balanced. Answer: Assume that if the useful estimation of two discretionary information values x and y are equivalent then x = y. Particular answer - B1: If f o g ( x ) = f o g ( y ), then x = y. How would you demonstrate B1? What is the key inquiry?

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The Key Question and Answer How would you indicate B1: If f o g ( x ) = f o g ( y ), then x = y. Answer: We take note of that B1 is of the structure If A`, the B`, and utilize the Forward-Backward technique to demonstrate the announcement If An and A`, then B`. ie., If the capacities f:X  X and g:X  X are both coordinated capacities and if f o g ( x ) = f o g ( y ), then x = y.

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So we start with B` : x = y and note that, since we don\'t know anything about x & y aside from that x & y are in the space X, we can\'t offer a sensible key conversation starter for B` so we ought to start the Forward Process for this new if-then proclamation.

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The Forward Process A`: The capacities f:X  X and g:X  X are both balanced capacities and f o g ( x ) = f o g ( y ) A`1: f(g(x)) = f(g(y)) (definition of arrangement) 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 demonstrated the announcement Now compose the confirmation.

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Write the Proof Statement Reason A: The capacities f:X  X Given and g:X  X are both balanced. A`: f o g ( x ) = f o g ( y ) Assumed to demonstrate f o g is 1-1 A`1: f(g(x)) = f(g(y)) meaning of piece A`2: g(x) = g(y) f is 1-1 by An A`3: x = y g is 1-1 by A B: f o g is 1-1 meaning of 1-1

Slide 23

Condensed Proof Suppose the f:X  X and g:X  X are both coordinated. To show f o g is balanced we accept f o g ( x ) = f o g ( y ). In this way f(g(x)) = f(g(y) and since f is balanced, g(x) = g(y). Since g is additionally balanced x = y. Subsequently f o g is coordinated. 

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