2010_lecture7 - CS245 - Winter 2010, Lecture 7 Shai...

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CS245 - Winter 2010, Lecture 7 Shai Ben-David 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 two-valued 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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2010_lecture7 - CS245 - Winter 2010, Lecture 7 Shai...

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