Question

2)Prove, using Boolean Algebra theorems, that the complement of XOR gate is XNOR gate(Hint : Prove that AB + AB = AB + ABby using De-Morgan’s theorem)3)Draw the K-Map for the following Boolean function. Obtain the simplified Sum of Products (SOP) expression, using the K-Map minimization procedure

.?(????)=∑?(1,2,3,5,7,9,11,13)

Answer 1

2) Let A,B be the Literals

F = A XOR B is A B = A'B+AB'

Compelment of the Function is

F' = (A'B+AB')'

F' = (A'B)' (AB')' { We know that (P+Q)'= P' Q' }

F' =  [(A')' +B'] [A' + (B')'] { We know that (PQ)'= P' + Q' }

F' =  [A +B'] [A' + B] { We know that (P')'= P }

F' =  A[A' + B] +B'[A' + B] { By Distributive law P(Q+R) = PQ+PR }

F' =  [AA' + AB] +[A'B' + BB']  { By Distributive law P(Q+R) = PQ+PR }

F' =  [0 + AB] +[A'B' + 0] { We know that PP'= 0 }

F' = AB + A'B' { We know that P+0= P }

F' = (A B)'

F' = A XNOR B

The complement of XOR gate is XNOR gate

3) Given Function is

F (a, b, c, d) =m (1, 2, 3, 5, 7, 9, 11, 13)

Above Function in K-map as follows

Simplified K-map as follows

The Simplified SOP of F (a, b, c, d) = a’b’c+a’d+b’d+c’d