notebook/notes/logic/classical/truth-tables.md

title	TARGET DECK	FILE TAGS	tags
Truth Tables	Obsidian::STEM	logic::classical	logic

Overview

In classical logic, every prop-logic can be written in disjunctive normal form (DNF) and conjunctive normal form (CNF). This is evident with the use of truth tables. To write a proposition in DNF, write its corresponding truth table and \lor each row that evaluates to T. To write the same proposition in CNF, apply \lor to each row that evaluates to F and negate it.

\neg (a \Rightarrow b) \Leftrightarrow c

It's truth table looks like

\begin{array}{c|c|c|c|c|c}
\neg & (a & \Rightarrow & b) & \Leftrightarrow & c \\
\hline
F & T & T & T & F & T \\
F & T & T & T & T & F \\
T & T & F & F & T & T \\
T & T & F & F & F & F \\
F & F & T & T & F & T \\
F & F & T & T & T & F \\
F & F & T & F & F & T \\
F & F & T & F & T & F
\end{array}$$

and it's DNF looks like

(a \land b \land \neg c) \lor (a \land \neg b \land c) \lor (\neg a \land b \land \neg c) \lor (\neg a \land \neg b \land \neg c)

It's CNF results from applying De Morgan's Law to the truth table's "complement":

\neg( (a \land b \land c) \lor (a \land \neg b \land \neg c) \lor (\neg a \land b \land c) \lor (\neg a \land \neg b \land c) )

%%ANKI
Basic
What construct is used to prove every proposition can be written in DNF or CNF?
Back: Truth tables.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707311868994-->
END%%

%%ANKI
Basic
Where are $\land$ and $\lor$ found within a DNF proposition?
Back: $\lor$ separates disjuncts containing $\land$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707311868998-->
END%%

%%ANKI
Basic
What is DNF an acronym for?
Back: **D**isjunctive **N**ormal **F**orm.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707311869000-->
END%%

%%ANKI
Basic
What is CNF an acronym for?
Back: **C**onjunctive **N**ormal **F**orm.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707311869002-->
END%%

%%ANKI
Basic
Where are $\land$ and $\lor$ found within a CNF proposition?
Back: $\land$ separates conjuncts containing $\lor$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707311869003-->
END%%

%%ANKI
Basic
What analog to truth tables is found in the algebra of sets?
Back: Membership tables.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716803633023-->
END%%

%%ANKI
Cloze
{Truth} tables are to propositions whereas {membership} tables are to set identities.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716803633029-->
END%%

%%ANKI
Basic
How many rows are in the truth table of identity $\neg (a \Rightarrow b) \Leftrightarrow c$?
Back: $2^3 = 8$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716803798112-->
END%%

%%ANKI
Basic
How many rows are in the membership table of identity $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$?
Back: $2^3 = 8$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716803798123-->
END%%

## Bibliography

* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).