Why sum of minterms or product of maxterms gives us the boolean function?

[Avichal]

Main Question or Discussion Point

I have thought about it and every-time I think I have an answer I try to explain it to myself and I fail. I want an intuition behind it and if there is a proof better.

Thank you

 

[Stephen Tashi]

You might get an answer if you state the question clearly.

 

[Avichal]

You might get an answer if you state the question clearly.

Oh okay, sorry if it was not clear
For example say two Boolean expressions x and y: its function is given
x y F
0 0 0
0 1 1
1 0 1
1 1 0

To find the function we need to add the minterms which are x'y and y'x. So function is x'y+y'x.
On what is minterms the link is here

But I don't understand why adding the minterms gives us the function.

 

[Stephen Tashi]

I'll suggest an intuitive way.

Visualize a Venn diagram where we have draw overlapping circles X and Y on a piece of paper. Smaller areas on the paper can be described by "coordinates" that tell whether the area is in-or-out of each set. So the possible coordinates in the descriptions are:

$X \cap Y$
$]X \cap Y^c$
$X^c \cap Y$
$X^c \cap y^c$

Any area that you can make using only some the above pieces can be written as a union of some of the pieces.

Of course you could draw an irregular area on the page that could not be described by the above procedure. For example, you could draw an area that was partly in $X \cup Y$ and partly out of $X \cup Y$. Such an area would not be "a function of" X and Y.

Returning to propositions, if a propositional function is a function of propositions x and y then it has a truth table. In the left columns of the table are listed all combinations of the truth and falsity values of x and y. The function is 1 precisely in the cases where the rows of its truth table make it 1. So writing the function as something like x'y + y'x amounts to saying the function is true on the rows of the truth table where entries of the leftmost two columns show the truth of x'y or y'x and it isn't true on any other rows.