11.6. Backward Reasoning

We have another proof mechanism called backward reasoning and this is an interesting proof mechanism. So imagine your goal is to prove a statement q to be true. The proof strategy that is involved here is instead of proving q to be true; we will try to find out statement p instead, which is true. Such that p → q is true.

Backward reasoning is a strategy used in mathematical proofs where the goal is to prove a specific statement, referred to as 'q.' Instead of directly proving 'q,' we first identify another statement 'p' that we know to be true, and we establish that 'p' implies 'q' (denoted as 'p → q'). If we can show that both 'p' is true and 'p' implies 'q,' then we can confidently conclude that 'q' must be true as well.

Think of backward reasoning like a detective solving a case. Instead of directly proving who committed the crime, the detective starts with a suspect (p) whose alibi checks out. The detective knows that if the suspect was at the crime scene (p), then the crime must have happened (q). Thus, by confirming the suspect's alibi, they indirectly prove the conclusion.

Let me demonstrate this proof mechanism by this theorem statement by proving this theorem statement we want to prove that for every distinct real numbers x and y their arithmetic mean is always greater than the geometric mean.

In applying backward reasoning to the statement about arithmetic and geometric means, we start with our goal (to prove that the arithmetic mean of two distinct numbers is greater than their geometric mean). We need to find a true statement (p). We argue that for the arithmetic mean to be greater than the geometric mean, a certain condition must hold. Through logical deductions, we identify that this condition will always be satisfied if x and y are distinct. Once we demonstrate that condition is true, we link it back to our original goal, proving that the arithmetic mean is indeed greater than the geometric mean.

Imagine you are comparing investments. You believe that one investment will yield higher returns than another. Instead of just presenting the numbers directly, you first establish a reliable indicator (the condition) showing that if the returns meet certain benchmarks, then your initial belief (the expected returns) will also hold true. This structured approach gives confidence to your conclusion.

So here is how we find the statement p. What we do is we keep on going backward starting with q, till we arrive at a statement p which is true and then we will try to see whether we can come back all the way from that p to our goal q.

In this chunk, we outline the systematic process of backward reasoning. First, we start with our conclusion (q) and identify necessary conditions that support it. We construct the logical pathway that connects these established facts back to our original aim. By ensuring each step maintains the truth from p to q, we validate our approach, eventually proving the main statement we set out to confirm.

This is similar to retracing your steps after forgetting where you parked your car. You think about the last place you remember being (your conclusion), then work backward to retrace your steps to other signs or landmarks that led you there. By confirming each step along the way, you ultimately find your car (prove q).

We start with the true statement p, we know that what all distinct x, y the square of their difference is always positive, from that we can come to the conclusion that the square of their sum is greater than 4 times xy.

In the final part of applying backward reasoning, we take p as a given—the distinctness of x and y ensures their difference is positive. This leads us logically to conclude that the square of their sum is greater than four times their product, which confirms our search for a true statement that supports our original claim about the arithmetic and geometric means. Following this logical progression allows us to establish the truth of the initial hypothesis (q).

Think about a cooking recipe where you need to prove that a dish tastes good (q). You first ensure that all ingredients are fresh and correctly measured (p). By confirming that fresh ingredients will result in a better dish, you can confidently claim that the dish will indeed taste great, once all steps from p lead naturally to q.

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