Propositional Logic Reference Sheet

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P
L
R
S
ROPOSITIONAL
OGIC
EFERENCE
HEET
 
Truth Tables for the Logical Operators
p ∧ q
p
q
p ∨ q
p → q
p
q
p
q
p
~p
T
T
T
T
T
T
T
F
T
T
T
T
F
F
T
F
T
F
T
T
F
F
F
T
F
F
T
T
F
T
T
F
F
F
F
F
F
F
F
T
Logical Equivalences
p ∧ q ≡ q ∧ p
1.
Commutative Laws:
p ∨ q ≡ q ∨ p
(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
2.
Associative Laws:
(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
3.
Distributive Laws:
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
p ∧ t ≡ p
4.
Identity Laws:
p ∨ c ≡ p
p ∨ ~p ≡ t
5.
Negation Laws:
p ∧ ~p ≡ c
~(~p) ≡ p
6.
Double Negation Law:
p ∧ p ≡ p
7.
Idempotent Laws:
p ∨ p ≡ p
p ∨ t ≡ t
8.
Universal Bound Laws:
p ∧ c ≡ c
~(p ∧ q) ≡ ~p ∨ ~q
9.
De Morgan’s Laws:
~(p ∨ q) ≡ ~p ∧ ~q
10.
Negations of t and c:
~t ≡ c
~c ≡ t
More Equivalences for Conditionals
Conditional written as “or” statement: p → q ≡ ~p ∨ q
(p → q) ∧ (q → p)
p ↔ q
Biconditional:
UIC CS 151
Prepared by D. Hogan referencing S. Epp’s Discrete Mathematics with Applications, 4th ed.

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