Every 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.
$$\neg (a \Rightarrow b) \Leftrightarrow c$$
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":
$$
\neg(
(a \land b \land c) \lor
(a \land \neg b \land \neg c) \lor
(\neg a \land b \land c) \lor
(\neg a \land \neg b \land c)
)
$$
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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.
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Basic
Where are $\land$ and $\lor$ found within a DNF proposition?
