How to know the boolean formula of a boolean function?

Question

Suppose A binary boolean function is showed by a true table. How can I know the (simplest) boolean formula which is interpreted by that function?

Answer 1

Answer by an example...

Given the following truth-table:

x | y | z | f
--|---|---|--
0 | 0 | 0 | 0
--|---|---|--
0 | 0 | 1 | 1
--|---|---|--
0 | 1 | 0 | 0
--|---|---|--
0 | 1 | 1 | 0
--|---|---|--
1 | 0 | 0 | 1
--|---|---|--
1 | 0 | 1 | 0
--|---|---|--
1 | 1 | 0 | 1
--|---|---|--
1 | 1 | 1 | 1

Your function is: $f(x,y,z)=(\neg{x}\wedge\neg{y}\wedge{z})\vee({x}\wedge\neg{y}\wedge\neg{z})\vee({x}\wedge{y}\wedge\neg{z})\vee({x}\wedge{y}\wedge{z})$

If the table contains more rows with $f=1$ than rows with $f=0$, then you can apply the above technique for the rows where $f=0$ instead, and then wrap the entire expression with a $\neg(\dots)$.