I was reading about Backward Reasoning, but I was not able to figure out how it works and what it really is.
While reading I came across the following example mentioned in Kenneth Rosen book.
Question: Given two positive real numbers x and y, their arithmetic mean is (x + y)/2 and their geometric mean is √xy. When we compare the arithmetic and geometric means of pairs of distinct positive real numbers, we find that the arithmetic mean is always greater than the geometric mean. [For example, when x = 4 and y = 6, we have 5 = (4 + 6)/2 > √4 · 6 =√24.] Can we prove that this inequality is always true?
Solution: To prove that (x + y)/2 > √xy when x and y are distinct positive real numbers, we can work backward. We construct a sequence
of equivalent inequalities. The equivalent inequalities are(x + y)/2 > √xy,
(x + y)2/4 > xy,
(x + y)2 > 4xy,
x2 + 2xy + y2 > 4xy,
x2 − 2xy + y2 > 0,
(x − y)2 > 0.
Because (x − y)2 > 0 when x ! = y, it follows that the
final inequality is true. Because all these inequalities are
equivalent, it follows that (x + y)/2 > √xy when x ! = y. Once we have
carried out this backward reasoning, we can easily reverse the steps
to construct a proof using forward reasoning. We now give this proof.Suppose that x and y are distinct positive real numbers. Then (x − y)2 > 0 because the square of a nonzero real number is positive. Because (x − y)2 = x2 – 2xy + y2 > 0. Adding 4xy to both sides, we obtain x2 + 2xy + y2 > 4xy. Because x2 + 2xy + y2 = (x + y)2, this means that (x + y)2 ≥ 4xy.
Dividing both sides of this equation by 4, we see that (x + y)2/4 > xy. Finally, taking square roots of both sides (which preserves the inequality because both sides are positive) yields (x + y)/2 > √xy. We conclude that if x and y are distinct positive real numbers, then their arithmetic mean (x + y)/2 is greater than their geometric mean √xy.
I didn't understand this proof. This proof is like if you have to prove a = b, then you add 5 both sides and get a+5 = b+5. Now using forward reasoning you prove a = b. Correct me if I am wrong.
Also, please illustrate how backward reasoning works.You can use appropriate examples.
Thank You.
Best Answer
Ordinarily, when proving something, you start with something you already know to be true, perform some steps which shows intermediate true things, and then reach a conclusion. In backwards reasoning you start from the thing you want to prove, and then look for what conclusions you can draw from it. While doing this, you try to make your steps reversible -- for example, from $x = y$ you can always conclude $0\cdot x = 0 \cdot y$, but that doesn't really help you because that step is not reversible.
Once you reach a "conclusion" that is obviously true, and every step in the proof is reversible, just reversing your backwards proof gives you an ordinary proof.
In my opinion, you should think of this less as a proof strategy and more as a strategy for exploring a problem.