Writing function using sigma?

Question

Suppose I was given:

$$g(w,x,y,z)=(x+ y ̅+ z ̅ )( w+ y ̅+ z ̅ )( w+ x ̅+ y ̅ )$$

Then I can write it like this:

\begin{align} g(w,x,y,z)&=(ww ̅+x+ y ̅+ z ̅ )( w+ xx ̅+y ̅+ z ̅ )( w+ x ̅+ y ̅ + zz ̅ )\\ &=(ww ̅+x+ y ̅+ z ̅ )( w+ xx ̅+y ̅+ z ̅ )( w+ x ̅+ y ̅ + zz ̅ )\\ &=(w+x+ y ̅+ z ̅ )\\ &\phantom{=\ }(w ̅+x+ y ̅+ z ̅ )\\ &\phantom{=\ }( w+ x+y ̅+ z ̅ )\\ &\phantom{=\ }( w+ x ̅+y ̅+ z ̅ )\\ &\phantom{=\ }( w+ x ̅+ y ̅ + z )\\ &\phantom{=\ }( w+ x ̅+ y ̅ + z ̅ ) \end{align}

From here how can I write the result using sigma?

It should be:

$$g(w,x,y,z)= ∏ (3,6,7,11)= ∑(0,1,2,4,5,8,9,10,12,13,14,15)$$

But I don't see how we concluded that.

• daniel: Note: I know I could do all of this using truth table but I remember that there was a shorter way after I simplified: $g(w,x,y,z)$ to contains all 4 inputs.
• Marcus Müller: Looking at your three equations: where exactly are you saving time compared to a truth table here?
• daniel: @MarcusMüller it may be that truth table takes the same time but I remember that there is another method for this
• Marcus Müller: you're right with that!
• Eugene Sh.: Note that your final expression has redundant (identical) terms. If you reduce them you will get the exact number of terms as in the Pi notation. After that take a close look at them to see the relation.

Answer 1

Formally this can be easily done using a truth table. But if you feel lazy to fill 16 rows, you can do it in your head in a bit shorter way.

Let's take a look at the original expression: $$g(w,x,y,z)=(x+ y ̅+ z ̅ )( w+ y ̅+ z ̅ )( w+ x ̅+ y ̅ )$$

Since it is POS, it will be more natural to work in the \$\Pi\$ notation and "inverted" logic.

So g will have the value of 0 whenever either of the three maxterms is zero. So let's start with first term: $$(x+ y ̅+ z ̅ )$$

When is it taking the value zero? It is when \$(xyz)=(011)\$ and the value of \$w\$ is "don't care". That is it is corresponding to the rows \0011ドル\$ and \1011ドル\$. That is the rows number \3ドル\$ and \11ドル\$. So these are going into the \$\Pi\$ representation. Then you repeat the same for the rest of the terms and complete the \$\Pi\$ representation. The \$\Sigma\$ representation is simply the complementary one (that is you fill the numbers out of the 0-15 range which do not appear under \$\Pi\$

P.S. If you wish you can add more formality to this method and first write down the explicit expansion of each maxterm into a canonical form as you did in the question (plus reducing the identical terms). Then you will have the exact same analysis as above, but for more terms without "don't cares". But I would say it is just extra work without extra benefit.