A general realization of a Boolean expression is shown in 12.15
Therefore, , which can be written as in the sum of product form. We arrive at this by looking at the combinations when the outout is one.
We can alternatively, express this in the product of sums form by looking at the combinatins when the outout is low as
Using SOP and POS, it can be implemented as follows:
Next, we will try to reduce the number of gates by combining terms suitably.
We can get the above by clubbing the s in the k-map shown.
Now, if we club the zeroes together in the k-map,
Check that we get the same expression by simplifying the product of sums expression (by using (X+Y)(X+Z)=X+YZ)