In certain application, four inputs A, B, C, D (both true and complement forms available)are fed to logic circuit, producing an output F which operates a relay. The relay turns on when F(ABCD)=1 for the following states of the inputs (ABCD):'0000', '0010' ,0101',0110','1101' and '1110'. States '1000' and '1001' do not occur, and for the remaining states, the relay is off. Minaining states, the relay is off. Minimize F with the help of a Karnaugh map and realize it using a minimum number of 3- input NAND gates.

Its is desired to generate the following three Boolean functions. $$\eqalign{ & {F_1} = a\,\overline b \,c + \overline a \,b\overline c + bc \cr & {F_2} = \,a\,\overline b \,c + ab + \overline a \,b\overline c \cr & {F_3} = \,\overline a \,\overline b \,\overline c + abc + \overline a c \cr} $$

By using an OR gate array as shown in figure where $${P_{1\,}}\,to\,{P_5}$$ are the product terms in one or more of the variables a, $$\overline a $$, b, $$\,\overline b $$, c and $$\overline c $$.