Note

This section is summarized based on the notes given by Han Si Yi

Logical Equivalences

Name	Formula
De Morgan's Laws	$\neg(P \lor Q) \equiv \neg P \land \neg Q$
De Morgan's Laws	$\neg(P \land Q) \equiv \neg P \lor \neg Q$
Implication	$P \to Q \equiv \neg P \lor Q$
Biconditional	$P \leftrightarrow Q \equiv (P \to Q) \land (P \leftarrow Q)$
Biconditional Alternative	$P \leftrightarrow Q \equiv (\neg P \lor Q) \land (P \lor \neg Q)$

CNF (Conjunctive Normal Form) Patterns

At Least K True

Condition	Formula
At least one from {a,b}	$a \lor b$
At least one from {a,b,c}	$a \lor b \lor c$
At least one from {a,b,c,d}	$a \lor b \lor c \lor d$
At least two from {a,b}	$a \land b$
At least two from {a,b,c}	$(a \lor b) \land (a \lor c) \land (b \lor c)$
At least two from {a,b,c,d}	$(a \lor b \lor c) \land (a \lor b \lor d) \land (a \lor c \lor d) \land (b \lor c \lor d)$
At least three from {a,b,c}	$a \land b \land c$
At least three from {a,b,c,d}	$(a \lor b) \land (a \lor c) \land (a \lor d) \land (b \lor c) \land (b \lor d) \land (c \lor d)$

At Most K True

Condition	Formula
At most one from {a,b}	$\neg a \lor \neg b$
At most one from {a,b,c}	$(\neg a \lor \neg b) \land (\neg a \lor \neg c) \land (\neg b \lor \neg c)$
At most one from {a,b,c,d}	$(\neg a \lor \neg b) \land (\neg a \lor \neg c) \land (\neg a \lor \neg d) \land (\neg b \lor \neg c) \land (\neg b \lor \neg d) \land (\neg c \lor \neg d)$
At most two from {a,b,c}	$\neg a \lor \neg b \lor \neg c$
At most two from {a,b,c,d}	$(\neg a \lor \neg b \lor \neg c) \land (\neg a \lor \neg b \lor \neg d) \land (\neg a \lor \neg c \lor \neg d) \land (\neg b \lor \neg c \lor \neg d)$
At most three from {a,b,c,d}	$\neg a \lor \neg b \lor \neg c \lor \neg d$

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