Forward and Backward Reasoning

Forward Reasoning

• You need a starting point for a proof
• Start from your premises, apply definitions, theorems, axioms, and make way towards the conclusions

Backward Reasoning

Start with the conclusion, and try to work backwards.

To prove $q$, we try to find some property $p$ that we can then use to prove $p\to q$

Example (Backward Reasoning): Given two positive real numbers $x$ and $y$, their arithmetic mean is $\frac{x+y}{2}$, and geometric mean is $\sqrt{xy}$. Prove that $\frac{x+y}{2}>\sqrt{xy}$ for two distinct positive real numbers

NOTE: THIS IS BACKWARD REASONING, IT IS NOT A PROOF

$$\begin{array}{r}\frac{a+b}{2}>\sqrt{ab}\\ \therefore <\text{Squaring Both Sides}>\\ (\frac{a+b}{2}{)}^2>ab\\ \therefore <algebra>\\ (\frac{a^2+2ab+b^2}{4}{)}^2>ab\\ \therefore <algebra>\\ (a^2+2ab+b^2)>4ab\\ \therefore <algebra>\\ (a^2-2ab+b^2)>0\\ \therefore <algebra>\\ (a-b{)}^2>0\end{array}$$