Find the complement of F (a, b, c, d) using Demorgan's Laws. Show the relevant reasoning.
F(a, b, c, d)=a + {(b + c).(b' + d')}

Given the function F (a, b, c) = Σ(0, 2, 3, 4, 6). Reduce it using Karnaugh's map.

A factory needs a minimum of 1200 tons of raw material and at least 100 workers to start its production. There are three suppliers each agreed to supply 600, 800 and 1250 tons of raw materials respectively.
A=1 if the first supplier supplies else it is 0.
B=1 if the second supplier supplies else it is 0.
C=1 if the third supplier supplies else it is 0.
D=1 if 100 workers are available else it is 0.
R=1 if production starts else it is 0.
(a) Taking A, B, C and D as inputs and R as output draw truth table for the problem stated above and derive its SOP expression.
(b) Reduce the above SOP expression using the K-map.

Given below is the truth table for a combinational circuit for which the input is a 3 bit number and output is its 2's complement.

Write SOP expression for the outputs P, Q and R. Reduce them, if possible, using the Karnaugh's map.