Boolean Algebra Proof 1 ABA BB B A 2 x xy z x z Solut

Boolean Algebra (Proof):

1) (AB)’(A’ + B)(B’ + B) = A’

2) x + xy + z = x + z  

Solution

1) (AB)’(A’ + B)(B’ + B) = A’

Note that (B’ + B) = 1 (complement law)

So (AB)’(A’ + B)(B’ + B) = (AB)’(A’ + B) = (A\' + B\')(A’ + B)

And (A\' + B\')(A\' + B) = A\' (See following truth table for verification)

Hence (AB)’(A’ + B)(B’ + B) = A’

2) x + xy + z = x + z

Taking x common,

x + xy + z

= x(1+y) + z

And or of 1 with any variable is 1 so , (1+y) = 1

= x.1 + z = x + z

Hence x + xy + z = x + z