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Unformatted text preview: 1 The Forward and Backward Method 1.1 Welcome Welcome to the second module in our course on proofs on the Forward and Backward method. Before going on, please do the assigned readings. 1.2 Introduction This lecture will consist of only a single example, but it will illustrate very clearly how mathe- maticians go forward from hypotheses and backwards from the conclusion to understand a proof. Consider the following statement. Proposition 1. If the right triangle XY Z with sides of length x and y and hypotenuse of length z has an area of z 2 / 4 , then the triangle XY Z is isosceles. Let’s discover a proof. The very first thing we need to do is identify the hypothesis and the conclusion. Our hypothesis is the statement: Hypothesis A : The right triangle XY Z with sides of length x and y and hypotenuse of length z has an area of z 2 / 4. This statement really has two parts Hypothesis H 1 : XY Z is a right triangle with sides of length x and y and hypotenuse of length z , and Hypothesis H 2 : XY Z has an area of z 2 / 4 The conclusion is the statement 1 Table 1: Hypothesis and Conclusion Statement Reason A : The right triangle XY Z with sides of length x and y and hypotenuse of length z has an area of Z 2 / 4 B : The triangle XY Z is isosceles. Table 2: Starting Statement Reason B : The triangle XY Z is isosceles. Conclusion B : The triangle XY Z is isosceles, and the whole proposition can be written briefly as A implies B . With confidence that we can discover a proof, let’s build a table with statements on the left and reasons on the right. Later on in the course, we will give interpretations, sometimes lengthy, of each statement but this proof is simple enough to give in a table. Our table will begin with the hypothesis and end with the conclusion. See Table 1. Since we do not yet know which part of the hypothesis we will use, and when we’ll use them, we’ll omit that statement until we do know. See Table 2. The plan is to work backwards from the conclusion and forwards from the hypothesis to construct a complete proof. The best way to work backwards is to ask questions. But not any questions....
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- Spring '08
- Pythagorean Theorem, Hypotenuse, triangle, triangle XY