Question

Use two different ways to prove X Y + Z = (X + Z)(Y + Z).
a) Use pure algebraic way

b) k-maps

Answer 1

a)

XY+Z=(X+Z)(Y+Z)

Applying distributive law to R.H.S(right hand side) then we get;

XY+Z= XY+XZ+ZY+ZZ

XY+Z= XY+XZ+ZY+Z(idempotents law , Z.Z=Z)

XY+Z=XY+Z(X+Y+1)

Applying property of 0 and 1 . that is 1+Z=1)

XY+Z=XY+Z.1

Applying property of 0 and 1, that is , Z.1=Z

so,

XY+Z=XY+Z

L.H.S=R.H.S

Hence proved

b)

Truth table of L.H.S , that is XY+Z is given below:

The k-map for the above truth table is given below:

We are finding SOP. We have obtained three groups

from group 1, we get,

001= X'Y'Z

011=X'YZ

So we get, X'Z

from group 2, we get,

001 = X'Y'Z

101= XY'Z

So we get Y'Z

from group 3, we get;

110=XYZ'

by combining all that we get L.H.S= X'Z+Y'Z+XYZ'

Truth table for the R.H.S is given below:

K-map for the truth table is given below:

we have obtained two groups

from group 1, we get ;

001= X'Y'Z

from the group 2 we get;

110=XYZ'

By combining them we get; R.H.S= X'Y'Z+XYZ'