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Logic Cheat Sheet
Prof. Woon PS 2703 August 27, 2007
Definitions
Valid argument Reasoning in which a conclusion follows necessarily from the premises presented, so that the conclusion cannot be false if the premises are true.
Statements Either true or false, but not both. Represented by letters.
Not (negation)
¬P
means “it is not the case that P”
And (conjunction)
P ∧ Q
means “both P and Q”
Or (disjunction)
P ∨ Q
means “either P or Q (or both)”
Conditional connective
P ⇒ Q
means
- “P implies Q”
- “if P then Q,”
- “P is sufficient for Q”
Converse
Q ⇒ P is the converse of P ⇒ Q
IMPORTANT! A conditional statement is NOT the same as its con- verse.
Contrapositive
¬Q ⇒ ¬P is the contrapositive of P ⇒ Q
A conditional statement IS EQUIVALENT to its contrapositive.
Biconditional connective
P ⇔ Q
means “P is necessary and sufficient for Q” or “P if and only if Q” (abbreviated iff)
Tautology A statement that is always true.
Contradiction A statement that is always false.
Logical equivalences
Double negation law
¬¬P ≡ P
Commutative laws
P ∧ Q ≡ Q ∧ P
P ∨ Q ≡ Q ∨ P
Associative laws
P ∧ (Q ∧ R) ≡ (P ∧ Q) ∧ R
P ∨ (Q ∨ R) ≡ (P ∨ Q) ∨ R
