Question

Draw the nonabbreviated logic diagram for Boolean expression (a'+b')((b'+c)+b'c) and prove it equals to b'+a'c. Give a reason for each step in your proof.

Answer 1

Logic diagram for Boolean expression F =  (a'+b')((b'+c)+b'c)

Given

F =  (a'+b')((b'+c)+b'c)

F =  ((b'+c) (a'+b')+b'c(a'+b'))  { By Distributive law P(Q+R) = PQ+PR }

F = ((b'+c) (b'+a')+ (a'b'c +b'b'c)) { By Distributive law P(Q+R) = PQ+PR }

F = ((b'+c) (b'+a')+ (a'b'c +b'c)) { We know that P'P'= P' }

F = ((b'+a'c)+ (a'b'c +b'c)) { By Distributive law (P+Q)(P+R) = P+QR }

F = (b'+a'c)+ b'c(a' +1)  { By Distributive law PQ+PR= P(Q+R) }

F = (b'+a'c)+ b'c(1) { We know that 1+P'=1 }

F = b'+ b'c +a'c

F = b'(1+ c) +a'c { By Distributive law PQ+PR= P(Q+R) }

F = b'(1) +a'c { We know that 1+P=1 }

F = b'+a'c

((b'+a'c)+ (a'b'c +b'c)) = b'+a'c

Hence Proved