Question

Digital Logic Design Lab

Prove the following Boolean Algebra theorems and properties by constructing Logic Circuits for each theorem/properties using our educational simulation software:

Q1-a) The Distributive Property:     a + ( b . c ) = ( a + b ) . ( a + c )

Q1-b) The Distributive Property:     a . ( b + c ) = ( a . b ) + ( a . c )

Answer 1

Distributive law :

a + (b.c) = (a + b)(a + c)

This is called OR distributes over AND.

Proof:

a + b.c = a * 1 + b.c → since a*1 = a

= a.(1 + b)+ b.c → since 1 + B = 1

= a * 1 + a.b + b.c

= a *(1 + c) + a.b + b.c → since a*a = a*1 = a

= a*(a + c) + b.(a + c)

= (a + c) (a + b)

a + b.c = (a + b) (a + c)

Hence proved.

2)

a.(b+c) = (a.b) + (a.c)

This is called AND distributes over OR.

Proof:
a.(b + c) = a.(b*1) + a.(c*1) → since 1 * b = b, 1 * c = c

= [(a.b)*(a*1)] + [(a.c) *(a*1)]

=[(a.b) * a] + [(a.c) *a]

= (a +1) (a.b + a.c)

= (a.b +a.c) → since 1 + a = 1

Hence proved.