State the principal of duality. Write the dual of:
(P + Q').R.1 = P.R + Q'.R

A combinational logic circuit with three inputs P,Q,R produces 1 if and only if an odd number of 0's are inputs.

1. Draw a truth table.
2. Derive a canonical SOP expression for the above truth table.
3. Find the complement of the above derived expression using De-Morgan's theorem and verify whether it is equivalent to its POS expression or not.

Minimize the expression using Boolean laws:
(A + B')(B + CD)'

In the following truth table x and y are inputs and B and D are outputs:

Answer the following questions:

(i) Write the SOP expression for D.
(ii) Write the POS expression for B