Binary Half Subtractor

Ask

For a binary half sub tractor having two inputs A and B, the correct set of logical expressions for the outputs D (= A minus B) and X (= borrow) are

(a)𝐷 = 𝐴  𝐵 + 𝐴̅  𝐵,   𝑋 = 𝐴̅ 𝐵

(b)𝐷 = 𝐴̅  𝐵 + 𝐴 𝐵̅ + 𝐴 𝐵̅, 𝑋 = 𝐴 𝐵̅

(c)𝐷 = 𝐴̅  𝐵 + 𝐴 𝐵̅,  𝑋 = 𝐴̅ 𝐵

                  (d)𝐷 = 𝐴  𝐵 + 𝐴̅ 𝐵̅,  𝑋 = 𝐴 𝐵̅

Answer is ( c )

Concept

For a binary half sub-tractor ,  please find the below truth table :

           To write down the Boolean expression for the same :
           D = f(A,B)= $f_{1} + f_{2}  \implies  $ D = 𝐴̅  𝐵 + 𝐴 𝐵̅,  
           X = f(A,B)= $f_{1} \implies $ 𝑋 = 𝐴̅ 𝐵