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CS245  Winter 2010, Lecture 7
Shai BenDavid
We have seen that a propositional formula can be viewed as a function from
{
T,F
}
n
to
{
T,F
}
(where
n
is the number of propositional variables in the
formula). Such functions, mapping vectors over a twovalued domain to two
values, are called
Boolean functions
(one should note that it does not really
mater how these values are denoted. Here we use the values
{
T,F
}
. In other
contexts one encounters Boolean functions over
{
0
,
1
}
or
{
1
,
1
}
). Any Boolean
function can be described by a table, listing all possible input vectors (as we
have already mentioned, there are 2
n
such vectors for a function over
n
Boolean
variables) and the values the function outputs on each of these inputs. In our
context, we call such tables ”truth tables”.
A very natural question to ask is: can propositions express
every
possible
Boolean function? In other words, given any truth table, does there exist a
proposition that has it as its truth table.
Theorem 1 (Post, 1921)
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 Spring '10
 Mr.Lushman

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