Logic and Proof Structures
Proofs Workshop
Notation p, q
propositions which may be true or false
¬p
not p it is not the case that p
p∧q
p and q are both true
p∨q
p is true or q is true, or both are true
p⇒q
if p p q p q
p⇔q
p is true if and only if q is true p iff q p is a necessary and sufficient condition for q.
p is true then q is true implies q only if q if p is a sufficient condition for q is a necessary condition for p
Truth Tables p
q
p∧q
p∨q
p⇒q
p⇔q
T T F F
T F T F
T F F F
T T T F
T F T T
T F F T
Logical Equivalence Let P and Q be logical expressions. Then P is logically equivalent to Q, P ≡ Q, if and only if P and Q have the same truth table. Methods of Proof Direct To prove p ⇒ q, assume p is true and deduce q must be true. Contrapositive To prove p ⇒ q, it is sufficient to prove ¬q ⇒ ¬p: assume q is false, deduce p must be false. ie (¬q ⇒ ¬p) ≡ (p ⇒ q)
Shirleen Stibbe
p
q
p⇒ q
¬q
¬p
¬q ⇒ ¬p
T T F F
T F T F
T F T T
F T F T
F F T T
T F T T
http://www.shirleenstibbe.co.uk