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I have been trying to solve this problem but I am unsure whether or not this is the right procedure. I have the following truth table.

Truth Table

I need to find the expression for F as a product of sums and as a sum of products. I also need to find the minimal sum of products.

If I find the product of sums, do I have to convert it directly to sum of products? Or will the sum of products directly from the truth table be equivalent?

So far (lowercase c denoting complement):

My product of sums: F = (X+Y+Z) (Xc+Y+Z) (Xc+Y+Zc)

My sum of products: F = (XcYcZ) + (XcYZc) + (XcYZ) + (XYZc) + (XYZ)

Now for my attempt at the simplification to get the minimal sum of products

F = (X+Y+Z) (Xc+Y+Z) (Xc+Y+Zc)

= (0 + XY + XZ + YXc + Y + YZ + ZXc + ZY + Z) (Xc+Y+Zc)

= (Y (X+Xc) + Z (X+Xc) + Y + YZ + Z) (Xc+Y+Zc)

= (Y + Z + YZ) (Xc+Y+Zc)

= (Y+Z) (Xc+Y+Zc)

= YXc + Y + YZc +ZXc +ZY + 0

= Y (ZC+Z) + YXc + Y + ZXc

= YXc + Y + ZXc

= Y + ZXc

Any advice/tips and help would be greatly appreciated. Thanks!

Brandon Chung
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  • "...Or will the sum of products directly from the truth table be equivalent?" They are certainly equivalent. I would expect to be easier (less steps) to find both expressions starting from the same one than getting one an then the other, from that one. It seem to me that what you call "Xc" is $X'$ or $X^c$, right? What about "o"? Is it $0$ (or equivalently, $F$, or $\bot$)? – amrsa Sep 30 '16 at 21:03

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