Reference Sheet for Logic and Program Proofs
Logical Equivalences
Definition of ∧Idempotent Laws DeMorgan’s Laws Distributive Laws
P∧ ¬P≡F alse p ∨p≡p¬(p∧q)≡ ¬p∨ ¬q p ∨(q∧r)≡(p∨q)∧(p∨r)
P∧F alse ≡F alse p ∧p≡p¬(p∨q)≡ ¬p∧ ¬q p ∧(q∨r)≡(p∧q)∨(p∧r)
P∧T rue ≡P
Definition of ∨Double Negation Absorption Laws Associative Laws
P∨ ¬P≡T rue ¬(¬p)≡p p ∨(p∧q)≡p(p∨q)∨r≡p∨(q∨r)
P∨F alse ≡P p ∧(p∨q)≡p(p∧q)∧r≡p∧(q∧r)
P∨T rue ≡T rue
Commutative Laws Implication Laws Biconditional Laws
p∨q≡q∨p p →q≡ ¬p∨q p ↔q≡(p→q)∧(q→p)
p∧q≡q∧p p →q≡ ¬q→ ¬p p ↔q≡ ¬q↔ ¬p
Inference Rules
Simplification Modus Ponens Modus Tollens Hypothetical Syllogism
p∧q p ¬q p →q
p→q p →q q →r
Therefore, p
Therefore, qTherefore, ¬pTherefore, p→r
Conjunction Addition Resolution Disjunctive Syllogism
p p p ∨q p ∨q
q¬p∨r¬p
Therefore, p∨q
Therefore, p∧qTherefore, q∨rTherefore, q
Universal Instantiation Universal Generalization Existential Instantiation Existential Generalization
∀xP (x)P(c)∃xP (x)P(c)
Therefore, P(c) Therefore, ∀xP(x) Therefore, P(c) Therefore, ∃xP (x)
Inference Rules For Program Proofs
Composition Rule Conditional Rule Conditional with Else Rule
p{S1}q(p∧condition){S}q(p∧condition){S1}q
q{S2}r(p∧ ¬condition)→q(p∧ ¬condition){S2}q
p{S1;S2}r p{if condition S}q p{if condition S1else S2}q
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