2E6 Digital  Problem Sheet 3
1.
Verify axioms A1 (b) to A5 (b) for the two-valued Boolean algebra.
2.
Prove by suitable means the following duals of theorems proved in the course notes:
T1(b)
X .X  X
T2(b)
X .0  0
T4(b)
X .(Y .Z )  ( X .Y ).Z
3.
T5(b)
( X .Y )  X   Y 
Ex 3.1(b)
X.(X+Y)=X
Using axioms, or theorems already proved, prove each of the following:
(i) X  X .Y  X  Y
(ii) X .( X   Y )  X .Y
(iii) X .Y  X .Z  Y .Z  X .Y  X .Z
(iv) ( X  Y ).( X   Z ).(Y  Z )  ( X  Y ).( X   Z )
4.
Simplify each of the following:
(i) A.B.C  A.B  A.B.C
(ii) X .Y .Z  X .Z
(iii) ( X  Y ).( X   Y )
(iv) A.C   A.B.C  A.C 
(v) ( B.C   A.D).( A.B   C.D )
F  X  Y .Z
find F  . Hence verify
F.F   0
and F  F   1
5.
If
6.
(a) Determine the logic function implemented by the circuit shown.
(b) Simplify the function and hence give an alternative logic circuit for the function.
C
X
A
B