Consider the following $$ER$$ diagram

The minimum number of table needed to represent $$M,$$ $$N,$$ $$P, R1, R2$$ is

In the Karnaugh map shown below, $$X$$ denotes a don’t care term. What is the minimal form of the function represented by the Karnaugh map?

If $$P, Q, R$$ are Boolean variables, then $$\left( {P + \overline Q } \right)$$ $$\left( {P.\overline Q + P.R} \right)\left( {\overline P .\overline R + \overline Q } \right)$$ Simplifies to

Given $${f_1},$$ $${f_3},$$ and $$f$$ in canonical sum of products form (in decimal) for the circuit. $$${f_1} = \sum {m\left( {4,5,6,7,8} \right)} $$$ $$${f_3} = \sum {m\left( {1,6,15} \right)} $$$ $$$f = \sum {m\left( {1,6,8,15} \right)} $$$
Then $${f_2}$$ is