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(Answered): [6 marks] Prove that each Theorem is true using the Postulatesand other Theorems of Boolean algebra ...
[6 marks] Prove that each Theorem is true using the Postulatesand other Theorems of Boolean algebra from below notes (note: youcannot use a Theorem to prove itself). a) xx = x
b) x + 1 = 1
Postulates and Theorems of Boolean Algebra o o o Postulate 1 (Identity): (a) x + 0 = x (b) x• 1 = x Postulate 2 (Complement): (a) x + x'= 1 (b) x•x'= 0 Postulate 3 (Commutative): (a) x + y = y + x (b) xy = yx Postulate 4 (Associative): (a) x+(y+z) = (x+y)+z (b) x(yz) = (xy)z º Postulate 5 (Distributive): (a) x(y+z) = xy + x2 (b) x + yz = (x+y)(x+z) o • Theorem 1 (Idempotency): (a) x + x = X (b) x.x = x • Theorem 2 (Null Elements): (a) x + 1 = 1 (b) x• 0 = 0 • Theorem 3 (Involution): (x')' = x Theorem 4 (DeMorgan's): (a) (x + y)' = x’y' (b) (xy)' = x' + y' Theorem 5 (Absorption): (a) x + xy = x (b) x(x + y) = x o o
