124 lines
3.7 KiB
Markdown
124 lines
3.7 KiB
Markdown
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---
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title: Truth Tables
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TARGET DECK: Obsidian::STEM
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FILE TAGS: logic::classical
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tags:
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- logic
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---
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## Overview
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In classical logic, every [[prop-logic|proposition]] 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.
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$$\neg (a \Rightarrow b) \Leftrightarrow c$$
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It's truth table looks like
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$$\begin{array}{c|c|c|c|c|c}
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\neg & (a & \Rightarrow & b) & \Leftrightarrow & c \\
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\hline
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F & T & T & T & F & T \\
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F & T & T & T & T & F \\
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T & T & F & F & T & T \\
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T & T & F & F & F & F \\
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F & F & T & T & F & T \\
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F & F & T & T & T & F \\
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F & F & T & F & F & T \\
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F & F & T & F & T & F
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\end{array}$$
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and it's DNF looks like
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$$
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(a \land b \land \neg c) \lor
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(a \land \neg b \land c) \lor
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(\neg a \land b \land \neg c) \lor
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(\neg a \land \neg b \land \neg c)
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$$
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It's CNF results from applying De Morgan's Law to the truth table's "complement":
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$$
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\neg(
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(a \land b \land c) \lor
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(a \land \neg b \land \neg c) \lor
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(\neg a \land b \land c) \lor
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(\neg a \land \neg b \land c)
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)
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$$
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%%ANKI
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Basic
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What construct is used to prove every proposition can be written in DNF or CNF?
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Back: Truth tables.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707311868994-->
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END%%
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%%ANKI
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Basic
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Where are $\land$ and $\lor$ found within a DNF proposition?
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Back: $\lor$ separates disjuncts containing $\land$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707311868998-->
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END%%
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%%ANKI
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Basic
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What is DNF an acronym for?
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Back: **D**isjunctive **N**ormal **F**orm.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707311869000-->
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END%%
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%%ANKI
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Basic
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What is CNF an acronym for?
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Back: **C**onjunctive **N**ormal **F**orm.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707311869002-->
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END%%
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%%ANKI
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Basic
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Where are $\land$ and $\lor$ found within a CNF proposition?
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Back: $\land$ separates conjuncts containing $\lor$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707311869003-->
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END%%
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%%ANKI
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Basic
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What analog to truth tables is found in the algebra of sets?
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Back: Membership tables.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716803633023-->
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END%%
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%%ANKI
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Cloze
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{Truth} tables are to propositions whereas {membership} tables are to set identities.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716803633029-->
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END%%
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%%ANKI
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Basic
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How many rows are in the truth table of identity $\neg (a \Rightarrow b) \Leftrightarrow c$?
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Back: $2^3 = 8$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1716803798112-->
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END%%
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%%ANKI
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Basic
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How many rows are in the membership table of identity $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$?
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Back: $2^3 = 8$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716803798123-->
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END%%
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## Bibliography
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* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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