Question

prove properties of Boolean algebr

4. Prove the following properties of Boolean algebras. Give a reason for each step. * (b) x + (x-y) = x x . (x + y) x (absorption properties) (c) (x y -x'x y)' -xy(DeMorgan's Laws) x +(y (xz))(x + y) (x (modular properties) (e) (x+y)·(x, + y) = y y+ y-y y)+x)-x+y (x-y) .(y+x') = x . y g x+y'-x+ y +x y)' (h) ((x . y) . z)+(y,z)=y.z

just A B and C please!

Answer 1

Remark Law Dual Pair AB-B+ A A.B- B.A Commutative Law A+B+C) = (A + B) + C A (BC) = (AB) C Associative Law A (B C) = AB + AC | A + (BC) = (A + B)(A + C) Distributive Law A.1 =A Identity Law/ Redundancy Law A+A =A A.A =A A·A = 0 A(A+B)-A A AB-A Absorption Law Involution Law A= A A(A+B)=A B+B=B

a) (x')' = x [Using Involution Law]

b)

x + (x . y) = x(1 + y) [Using Distributive Law]

= x [Using Identity Laws]

c)

(x + y)' = x' . y' [Using DeMorgan's Law]

(x . y)' = (x' + y') [Using DeMorgan's Law]