Negation

Question 1

Universal statements (involving “all”, “ever”, or “for all”)

Answer 1

In general, negate: “For all x, P(x)” becomes “There exists an X such that ¬P(x)

Question 2

Existential Statements (involving “there exists”)

Answer 2

In general, negate: “There exists an x such that P(x)” becomes “For all x, ¬P(x).”

Question 3

Conditional Statements (”if…then…”)

Answer 3

In general, negate: “If P, then Q” becomes “P and not Q”

Question 4

Quantifier-Free Statements

Answer 4

For statements without quantifiers (like P∧Q, P∨Q, etc), use De Morgan’s Laws
- The negation of P∧Q (both P and Q are true) is ¬P∨¬Q (at least one of P or Q is false)
- The negation of P∨Q (at least one of P or Q is true) is ¬P∧¬Q (both P and Q are false)