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
Proof by contradiction To prove p, it is sufficient to prove ¬p ⇒ false: assume p is false and deduce a contradiction (falsehood). ie (¬p ⇒ false) ≡ p p
¬p
false
¬p ⇒ false
T F
F T
F F
T F
Proof of a biconditional: Method 1: Prove p ⇒ q and q ⇒ p, ie (p ⇔ q) ≡ (p ⇒ q) ∧ (q ⇒ p) p
q
p⇔ q
p⇒q
q⇒p
(p ⇒ q) ∧ (q ⇒ p)
T T F F
T F T F
T F F T
T F T T
T T F T
T F F T
Alternatively: (p ⇒ q) ∧ (¬p ⇒ ¬q) or (q ⇒ p) ∧ (¬q ⇒ ¬p) using contrapositives. Method 2: Prove that both are true, or neither is true, ie (p ⇔ q) ≡ (p ∧ q) ∨ (¬p ∧ ¬q) p
q
p⇔ q
¬p
¬q
p∧q
¬p ∧ ¬q
(p ∧ q) ∨ (¬p ∧ ¬q)
T T F F
T F T F
T F F T
F F T T
F T F T
T F F F
F F F T
T F F T
Shirleen Stibbe
http://www.shirleenstibbe.co.uk