Mathematical Programming

Positivity

What if we wanted to make sure our input was positive before we used it in a function?

To our rescue comes the absolute value function. By dividing the number by its magnitude, we can determine its direction.

∣x∣x⇒{+→1,−→−1}\frac{\left|x\right|}{x}\Rightarrow\left\{+\rightarrow1, -\rightarrow-1\right\}

If positive, we'll get 1,1, and if negative, we will get −1.-1.

Now, we need to configure the positive-negative function to output binary, so that we can use it as a building block later. This means outputting 11 if it is positive, and 00 if it is negative.  

isPositive(x)=∣x∣x+1⇒{+→2,−→0}\text{isPositive}\left(x\right) = \frac{\left|x\right|}{x}+1\Rightarrow\left\{+\rightarrow2, -\rightarrow0\right\}
isPositive(x−y)=∣x∣x+12⇒{+→1,−→0}\text{isPositive}\left(x-y\right) = \frac{\frac{\left|x\right|}{x}+1}{2}\Rightarrow\left\{+\rightarrow1, -\rightarrow 0 \right\}

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