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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. screenshot from linked to power point presentation

Celeritas
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1 Answers1

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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 $\sum(1,2,4,7)$)

JMoravitz
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  • Well that sucks. So every time I see $\sum(a,b,...c)$ I am to assume the truth table was written in Lexicographical Order, or is there a different way to interpret it? – Celeritas Sep 28 '15 at 04:50
  • @Celeritas unfortunately, yes, it seems that is the author's intent. It is not that hard of a habit to get into though, and when thinking in terms of binary it should make sense, what with it corresponding to simply listing the numbers in base 2 in order. $0000,0001,0010,0011,0100,0101,\dots$. I don't do any work with these sorts of representations, so I'm not sure how common they are, but that seems to be how to make sense of what is written. – JMoravitz Sep 28 '15 at 04:58