3 Draw a Truth Table using the following C Prove the follow

#3

Draw a Truth Table using the following + C) Prove the following theorems algebraically i) (A + B)(A + B) = A ii) XY + XY\' = X iii) X + XYZ = X iv) X(X + Y) = XY Simplify each of the following expression theorems. State the theorems used. a) X\'Y\'Z + (X\'Y\'Z) b) (A\'B + C + D)(A\'B + D) Draw a circuit using the following expression expression) F =- (AB + C\'D)(B\' + AC\')

Solution

a) (A+B)(A+B\')=A

(A+B)(A+B\')

=> AA + AB\' + AB + BB\'

=> A+AB\'+AB + BB’ {by Idempotent law: AA=A }

=> A+AB\'+AB + BB’ {by Complementary law: BB\' =0}

=> A(1+B\'+B) {by Absorption law}

=> A.1 {by Identity law}

=>A

b) XY + XY’= X

XY + XY’

=> X( Y + Y’ ) {by Distributive law}

=> X   {by Complementary law: Y+Y’ =1}

c) X + XY = X

X + XY

=> X(1+Y) {by Distributive law}

=> X.1   { by Identity law: A.1=A }

=>X

d) X(X\' + Y) = XY

X(X\' + Y)

=> XX\' + XY {by Distributive law}

=> 0 + XY {by Complementary law: BB\' =0}

=> XY { by Identity law: A + 0 =A }