I'm trying to understand what is meant by the notation $F(x,y,z)=\sum m(0,1,2,3)$
I found this webpage but it's still unclear to me. It states the function F is defined by the truth table
$A$ $B$ $C$ $F$
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 0
1 1 0 0
1 1 1 1
and that "A compact notation is to write only the numbers of the minterms included in F" which is $\sum (1,2,4,7)$ I don't understand this because the rows in the truth table are interchangeable; 0,0,0,0 could be the last row instead of the first. Also, it appears the first row is starting from 0, not 1?
Could someone please explain this notation to me and what is it called?
Another example comes from slide 11 of this power point presentation.
Best Answer
Although, yes, in practice it doesn't usually matter what order you write a truth table, this author seems to tacitly be making the assumption that you must write a truth table in Lexicographical Order. I.e, flip the furthest bit first.
Thought of in a different way, if you interpret the entries as a binary number, they are written from smallest to largest. $000_2 = 0, ~001_2=1, ~010_2=2, ~011_2=3,~\dots$
By making sure such truth tables are written in a very specific order, the representation the author refers to is then well-defined and unique.
Thinking of a boolean function on three variables as a function from $\mathbb{F}_2^3\to \mathbb{F}_2$, their notation $F=\sum(1,2,4,7)$ is just saying that $F(0,0,0)=0,~F(0,0,1)=1,~F(0,1,0)=1,~F(0,1,1)=0,~F(1,0,0)=1,~F(1,0,1)=0,~F(1,1,0)=0,~F(1,1,1)=1$ by simply only listing the ones with an output of one in shorthand notation based on where they appeared on the list. ($F(1,1,1)$ for example being the seventh entry on the list, where we start counting from zero, hence the $7$ in $\sum(1,2,4,7)$)