4.1 16.4 years
4.2 Prove: XY + YZ + /XZ = XY + /XZ
Multiply through by 1 in the form ( A + /A )
XY(Z + /Z) + YZ(X + /X) + /XZ( Y + /Y ) = XY + /XZ
XYZ + XY/Z + XYZ + /XYZ + /XYZ + /Y/XZ = XY + /XZ
Get rid of common terms
XY/Z + XYZ + /XYZ + /X/YZ = XY + /XZ
Factor out 1 again
XY(Z + /Z) + /XZ(Y + /Y) = XY + /XZ
And you get:
XY + /XZ = XY + /XZ
4.3 Any function can be reduced to a 2 level circuit. PLAs can already implement any 2 level circuit. Therefore, he lies and you shouldn't give the guy any money.
4.4 Simplify:
fg + (/c + /b)ad(e + g) + ad/e/g + f(/b + /c) + abcd + bcf
Gather the terms that have f in them and the terms that have ad in them.
f(g + /b + /c + bc) + ad(e/c + g/c + /be + /bg + /e/g + cb )
Note that (/b + /c + bc) == 1. so f(g + /b + /c + bc) becomes f.
Also note that (/ce + /cg + /e/g) == (/c + /e/g) (Try figuring this out using K-maps ), and (/be + /bg + /e/g) == (/b + /e/g) which leaves:
f + ad( /b + /c + /e/g + cb )
Note that ( /b + /c + cb ) == 1 so the final simplified form is:
f + ad.
The answer is: 2 gates.
