Negation of Statements and Quantifiers

Pushing a negation through a conjunction or disjunction flips the connective and negates each part:

• $\sim(p \wedge q) \equiv \;\sim p \vee \sim q$
• $\sim(p \vee q) \equiv \;\sim p \wedge \sim q$

Apply them outermost first, then repeat on whatever is left inside. A double negation cancels: $\sim(\sim p) \equiv p$.

$\sim(p \to q) \equiv p \wedge \sim q$.

• The result is a conjunction, because the one false row of $p \to q$ is $p = T,\; q = F$.
• The arrow disappears entirely. Any option that still contains an arrow is wrong by shape alone.
• For a nested conditional, push the negation in one layer at a time.

$\sim(p \leftrightarrow q) \equiv (p \wedge \sim q) \vee (\sim p \wedge q)$, read as 'exactly one of p and q is true'.

• A useful companion: $p \leftrightarrow \sim q$ is itself equivalent to $\sim(p \leftrightarrow q)$.
• Consequently $\sim(p \leftrightarrow \sim q) \equiv p \leftrightarrow q$ — a negation and an internal negation cancel each other out.

In a verbal stem, 'if and only if' is the signal; the negation always reads as one part holding without the other.

For a verbal compound statement:

• Assign letters to the simple statements, keeping each one positive.
• Write the symbolic form, watching for hidden connectives: 'but' is AND, 'unless' is usually a conditional, 'if and only if' is a biconditional.
• Negate symbolically using De Morgan and the conditional and biconditional rules.
• Translate back, and check that every connective in your answer flipped.

Negation swaps the quantifier and negates the predicate:

• $\sim[\forall x,\, p(x)] \equiv \exists x \text{ such that } \sim p(x)$
• $\sim[\exists x \text{ such that } p(x)] \equiv \forall x,\, \sim p(x)$

With several quantifiers, flip them in order, left to right, and negate the predicate at the end. An inequality inside the predicate reverses and loses or gains its equality: the negation of $x \geq M$ is $x < M$.