Question

3. a) Simplify the expression (ab + a’b’)(cd + c’d’) + (ab)’

b). Prove that the expression x’y XOR xy’ = x XOR y is true.

c). Simplify the function f(x, y, z) = xyz’ + xy’z’ + x’y as much as possible and give the CPOS and CSOP of f

Answer 1

(ab + a’b’)(cd + c’d’) + (ab)’
(ab + a’b’)(cd + c’d’) + (a'+b')
[ab(cd + c’d’) + a’b’(cd + c’d’)] + (a'+b')
[(abcd + abc’d’) + (a’b’cd + a’b’c’d’)] + (a'+b')
abcd + abc’d’ + a’b’cd + a’b’c’d’ + a'+b'

The Simplified Expression is F = a’+b'+cd+c'd'   

b) Given

x’y XOR xy’ = x XOR y
Take L.H.S
x’y XOR xy’
(x’y)'xy’+x’y(xy’)'
[(x’)'+y']xy’+x’y[x'+(y’)']
[x+y']xy’+x’y[x'+y]
[xxy’+xy’y']+[x'x’y+x’yy]
[xy’+xy']+[x'y+x'y]
[xy']+[x'y]
[xy']+[x'y]
x XOR y

= R.H.S

x’y XOR xy’ = x XOR y

c) f(x, y, z) = xyz’ + xy’z’ + x’y
= xz’(y + y') + x’y
= xz’(1) + x’y
= xz’ + x’y

The Simplified SOP is f(x, y, z) = xz’ + x’y

The Simplified POS is f(x, y, z) = (x’+z’)(x+y)